math.kangwon.ac.krmath.kangwon.ac.kr/~yhpark/dvis/halgebra.pdf · 10 j ] 1 © ç h v 1.2 â k ) â...
TRANSCRIPT
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V� 1 *�× !B� 1
]j 1.1 ]X�ç�H_� &ñ_� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
]j 1.2 ]X�ï�r1lx+þA�<Êú . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
]j 1.3 ]X�ÂÒì�rç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
V� 2 *�× 'K�».É!B�, n�».É!B�, �ßjÝ~!B� 19
]j 2.1 ]X�í�H8�ç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
]j 2.2 ]X�u�8�ç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
]j 2.3 ]X�'��§>=ç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
V� 3 *�× �b�#bÇÚÿ? �b�#b!B� 31
]j 3.1 ]X�eç#�ÀÓü< Lagrange &ño� . . . . . . . . . . . . . . . . . . . . . . . . . 32
]j 3.2 ]X�&ñ½©ÂÒì�rç�Hõ� eç#�ç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
V� 4 *�× ËÂø5��»jÅ]���G�f!B� 43
]j 4.1 ]X�f��&h� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
]j 4.2 ]X�Ä»ô�ÇÒqt$í��6\�ç�H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
V� 5 *�× !B��+ ¼ÇУ� ø� Sylow Ça�h� 47
]j 5.1 ]X�ç�H_� ���6 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
]j 5.2 ]X�Sylow &ño� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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V� 1 *�×
!B�
1
2 ]j 1 �©� ç�H
V� 1.1 â� !B��+ Ça��+
ô�Ç |9�½+Ë S 0A\�"f l�/�×¥o>ñ5Ñ(binary operation)s�êøÍ
∗ : S × S → S
(a, b) 7→ ∗(a, b)
��� �<Êú\�¦ ú� � 9 s� M: (S, ∗)\�¦ l�/�×Ä©�¿(binary structure)�� ô�Ç��. s��½Ó���íß� ∗_�(a, b)\�"f_� �<Êú°ú�¦ �âĺ\� ����
a ∗ b, ab, a+ b, a ◦ b, a× b, a#b
1pxܼ�Ð æ¼�� ÅÒ�Ð ���íß��¦ Òqt|ÄÌ �#� ab�Ð ��H��.�� ���×e� 1.1.1 (1) R\�"f &ñ_��)a +, ×, −��H s��½Ó���íß�s��� ÷��H s��½Ó���íß�s� ��m���.
(2) N0A\�"f &ñ_��)a +, ×��H s��½Ó���íß�s��� −, ÷�Ér s��½Ó���íß�s� ��m���.�� ��K±ÓÞDº 1.1.2 (1) ëß���� e��_�_� a, b, c ∈ S \� @/K�
a ∗ (b ∗ c) = (a ∗ b) ∗ c
s�$íwn� ���� ∗\�¦ÚrTÒ¼�\�(associative)s��� � 9���½+Ë&h�s��½Ó½�\�¦ð5�!B�(semigroup)s��� ô�Ç��.
(2) ëß���� e��_�_� a, b ∈ S \� @/K�a ∗ b = b ∗ a
�� $íwn� ���� ∗\�¦ ��».É�\�(commutative)s��� ô�Ç��. ��8�s��½Ó½�_� �âĺ ���íß��¦M:M:�Ð a+ b�Ð æ¼�¦, Á' û�gl�/�×Ä©�¿���¦� ô�Ç��.
(S, ∗)\�¦ ���½+Ë&h� s��½Ó���íß�s��� ��¦ a1, a2, . . . , an ∈ S�� ���. a1, a2, . . . , an�¦ ÅÒ#Q��� í�H"f@/�Ð Y�L ���H ~½ÓZO��Ér e��_�_� m < n\� @/K� (a1 · · · am)(am+1 · · · an) 1px #��Q��t��� e����. ³ðï�r&h���� Y�L�Ér ) ±ú�&h�ܼ�Ð
a1 · · · an−1an = (a1 · · · an−1)an
ܼ�Ð &ñ_�ô�Ç��. Õª�Q��� #Q*�ô�Ç Y�L� �¿º ³ðï�r&h���� Y�Lõ� °ú 6£§�¦ �Ð{9� ú e����. Õª�QÙ¼�Ð F�c ñ\�¦ Òqt|ÄÌ ��¦ a1a2 · · · ans����¦ +�� Áº~½Ó ���.
]j 1.1 ]X� ç�H_� &ñ_� 3
�� ��K±ÓÞDº 1.1.3 (1) ô�Ç "é¶�è e ∈ S�� �>rF�K�"f ���H x ∈ S \� @/K�x ∗ e = e ∗ x = x
�� $íwn� ���� e\�¦ /�×�� xjS(identity, unity)�� � 9 �½Ó1px"é¶�¦ ����� ìøÍç�H�¦ �¿�¿l�Þ«���¦ ô�Ç��.
(2) ��s�×¼ (S, ∗)_� "é¶�è a\� @/K� #Q�"� "é¶�è a′ ∈ S�� �>rF�K�"fa ∗ a′ = a′ ∗ a = e
��$íwn� ���� a′�¦ a_�%i�"é¶s��� ��¦ a−1�Ð��H��.%i�"é¶�¦°ú���H"é¶�è\�¦í5�xjS(unit)s��� ô�Ç��. »!lr½�_� �âĺ\���H �Ð:�x a_� %i�"é¶�¦ −a�Ð ��H��. a�� éß�"é¶s����, a−1�éß�"é¶s� 9 &ñ_�\� _�K� (a−1)−1 = as���.
(S, ∗)�� ��s�×¼s��¦ a ∈ Ss���� a0\�¦ �½Ó1px"é¶s��� &ñ_� ��¦ e��_�_� �����ú n\� @/K�
an =n︷ ︸︸ ︷
a · · · a
���¦ &ñ_�ô�Ç��. ëß���� a�� %i�"é¶�¦ ��t����
a−n =
n︷ ︸︸ ︷a−1 · · · a−1
���¦ &ñ_�ô�Ç��. Õª�Q���an+m = anam, (am)n = amn
õ� °ú �Ér t�úZO�gË:s� $íwn�ô�Ç��. »!lr½�_� �âĺ��H an�¦ na�Ð ��H��.�� ���×e� 1.1.4 (N,+)��H ìøÍç�Hs��¦ (Z,×)��H ��8���s�×¼s���.�� ���¿Ça�h� 1.1.5 (S, ∗)\�¦ ìøÍç�Hs��� ���.
(1) ëß���� �½Ó1px"é¶s� �>rF� ���� Õª��Ér Ä»{9� ���.
(2) ëß���� a_� %i�"é¶s� �>rF� ���� Õª��Ér Ä»{9� ���.
7£x"î. e, e′\�¦ �½Ó1px"é¶s��� ���� e = e ∗ e′ = e′. ¢ô�Ç a′, a′′s� a_� %i�"é¶s����a′ = a′ ∗ e = a′ ∗ (a ∗ a′′) = (a′ ∗ a) ∗ a′′ = a′′.
���H "é¶�è�� %i�"é¶�¦ ��t���H ��s�×¼\�¦ ç�Hs��� ô�Ç��. 7£¤,
4 ]j 1 �©� ç�H�� ��K±ÓÞDº 1.1.6 ��6£§_� �|��¦ëß�7ᤠ���Hs��½Ó½� (G, ∗)\�¦, ¢��Hçß�éß�y� G\�¦!B�(group)s��� ô�Ç��.
(G1) (���½+Ë&h�) ∗�� ���½+Ë&h�s���.
(G2) (�½Ó1px"é¶) ô�Ç "é¶�è e ∈ G�� �>rF�K�"f ���H a ∈ G\� @/K� e ∗ a = a ∗ e = a�� $íwn�ô�Ç��.
(G3) (%i�"é¶) y�� a ∈ G���� a′ ∈ G�� �>rF�K�"f a ∗ a′ = a′ ∗ a = e�� $íwn�ô�Ç��.
G�� ç�Hs���� e��_�_� a, b, c ∈ G\� @/K� ��6£§s� $íwn�ô�Ç��.
a ∗ b = a ∗ c⇒ b = c. (ýa�è��ZO�gË:) (1.1)
b ∗ a = c ∗ a⇒ b = c. (ĺ�è��ZO�gË:) (1.2)
Õª�QÙ¼�Ð e��_�_� g ∈ G\� @/K� ĺ8£¤Y�L!lr �<Êúρg : G→ G, a 7→ ag
ü< ýa8£¤Y�L!lr �<Êúλg : G→ G a 7→ ga
��H ���éß����<Êús���.
��8���� ç�H�¦ ��».É!B�(commutative group) ¢��H ��G�f!B�(abelian group)s����¦ ô�Ç��.
G�� ��6\�ç�Hs��¦ a, b ∈ Gs���� e��_�_� &ñú n\� @/K� (ab)n = anbns� $íwn�ô�Ç��.�� ���×e� 1.1.7 ç�H G_� ���H "é¶�è g ∈ G\� @/K� g2 = es���� G��H ��6\�ç�Hs���.
7£x"î. e��_�_� a, b ∈ G\� @/K� e = (ab)2 = ababs�Ù¼�Ðb−1a−1 = ab
s���. ժ���X<e = a2 = aa, e = b2 = bb
\�"f a−1 = a, b−1 = bs�Ù¼�Ð ba = abs���.�� ���×e� 1.1.8 n ∈ Z\�¦ �¦&ñ ��¦ nZ = {nk | k ∈ Z}�� ���� (nZ,+)��H �½Ó1px"é¶s� 0��� ��6\�ç�Hs����� ���×e� 1.1.9 (Q,+), (R,+), (C,+) �¿º �½Ó1px"é¶s� 0��� ��6\�ç�Hs���.
]j 1.1 ]X� ç�H_� &ñ_� 5
�� ���×e� 1.1.10 Q+, R+\�¦ y��y�� �ª�_� Ä»o�ú, z�ú�� ���� s�[þt�Ér Y�L!lr\� @/K� ��6\�ç�H�¦ s�ê�r��.�� ���×e� 1.1.11 m ∈ Z\�¦ �¦&ñ ��¦
Q[√m] = {a+ b
√m | a, b ∈ Q}
�� ���� Q[√m]�Ér »!lr\� @/K� ��6\�ç�H�¦ s�ê�r��.�� ���×e� 1.1.12 Q∗, R∗, C∗\�¦ y��y�� 0�¦ ]jü@ô�Ç Ä»o�ú, z�ú, 4�¤�èú_� |9�½+Ës��� ½+É M:,
(Q∗,×), (R∗,×), (C∗,×)�Ér �½Ó1px"é¶s� 1��� ��6\�ç�Hs���.�� ���×e� 1.1.13 F\�¦ Q,R,C ×�æ ����� ��¦ n,m�¦ �ª�_� &ñú�� ���. F_� "é¶�è\�¦ �½Óܼ�Ð °ú���H ���H n×m '��§>=[þt_� |9�½+Ë Mn×m(F )�Ér '��§>=_� »!lr\� @/K� ç�H�¦ s�ÀÒ�¦
GLn(F ) = {g ∈Mn×n(F ) | det g 6= 0}
ü<SLn(F ) = {g ∈Mn×n(F ) | det g = 1}
�Ér '��§>=_� Y�L!lr\� @/K� ç�H�¦ s�ê�r��. ¢ F0A\�"f_� 7�'�/BNçß�� 7�'�½+Ë\� @/K� ç�H�¦ s�ê�r��.�� ���×e� 1.1.14 �¦&ñ�)a �����ú n\� @/K�
Un = {z ∈ C | zn = 1} = {e2πk/n | 0 ≤ k ≤ n− 1}
�Ér Y�L!lr\� @/K� ç�H�¦ s�ê�r��. ¢C1 = {z ∈ C | |z| = 1}
� Y�L!lr\� @/K� ç�H�¦ s�ê�r��.�� ���×e� 1.1.15 Zn = {0, 1, . . . , n− 1}s��� ���. Zn_� ¿º "é¶�è a, b\� @/K� ��Ðüw!lr &ño�\�¦ æ¼���
a+ b = qn+ c, (0 ≤ c < n)
s� $íwn� ���H &ñú q, c�� �>rF� ���HX<, s� M:a+n b = c,
6 ]j 1 �©� ç�H
�� &ñ_� ���� (Zn,+n)�Ér ç�Hs���. ¢ab = qn+ d, (0 ≤ d < n)
s� $íwn� ���H &ñú q, d�� �>rF� ���HX<, s� M:a×n b = d,
��&ñ_� ���.Õª�Q��� ax ≡ 1 (mod n)s�Ä»{9�ô�ÇK�\�¦°ú��¦�9�¹Ø�æì�r�|��Ér (a, n) = 1s�Ù¼�Ð
Z∗n = {a ∈ Zn | (a, n) = 1}
s����¦ ¿º��� (Z∗n,×n)� ç�Hs� �)a��. l� ñ_� ¼#�_��©� a\�¦ Õªzª� a�Ð æ¼ 9 Z∗
n_� "é¶�è_� >hú\�¦ φ(n)s��� ��H��.�� ���×e� 1.1.16 R\�"f R�Ð ����H ���H (���5Åq���) �<Êú[þt_� |9�½+Ë F (R) (C(R))õ� p�ì�r��0pxô�Ç �<Êú[þt_� |9�½+Ë D(R)�Ér �<Êú_� »!lr\� @/K� ç�H�¦ s�ê�r��.�� ���×e� 1.1.17 X 6= {}\�¦ e��_�_� |9�½+Ës��� ���.
SX = {f : X → X | f��H ���éß����<Êú}�� ���� �<Êú_� ½+Ë$í�¦ s��½Ó���íß�ܼ�Ð �#� SX��H ç�Hs� �)a��. s� ç�H (SX , ◦)�¦ X0A_�7��b�!B�(symmetric group, permutaion group)s��� � 9 :£¤y� X = {1, 2, . . . , n}{9� M:,
SX\�¦ n��+ n�».É!B� ¢��H @/g�Aç�Hs��� � 9 çß�éß�y� Snܼ�Ð �����·p��.�� ���×e� 1.1.18 G1, G2, . . . , Gn�¦ ç�Hs��� ���. Y�L|9�½+ËG1 ×G2 × · · · ×Gn = {(a1, a2, . . . , an) | ai ∈ Gi} (1.3)
�Ér ���íß�(a1, a2, . . . , an)(b1, b2, . . . , bn) = (a1b1, a2b2, . . . , anbn)
\�@/K�ç�H�¦s�ê�r��.z�]j�Ð ei\�¦Gi_��½Ó1px"é¶s��� ����s�ç�H_��½Ó1px"é¶�Ér (e1, e2, . . . , en)
s��¦(a1, a2, . . . , an)−1 = (a−1
1 , a−12 , . . . , a−1
n )
s���. s� ç�H�¦ G1, G2, . . . , Gn_� ÒÏ��\�!B�(direct product)���¦ ô�Ç��.�� ���¿Ça�h� 1.1.19 ���½+Ë&h� ìøÍç�H G\� @/K� ��6£§s� $íwn� ���� G��H ç�Hs���.
]j 1.1 ]X� ç�H_� &ñ_� 7
(1) (ýa8£¤�½Ó1px"é¶) ô�Ç "é¶�è e ∈ G�� �>rF�K�"f ���H a ∈ G\� @/K� e ∗ a = a�� $íwn�ô�Ç��.
(2) (ýa8£¤%i�"é¶) y�� a ∈ G���� a′ ∈ G�� �>rF�K�"f a′ ∗ a = e�� $íwn�ô�Ç��.
7£x"î. e��_�_� a ∈ G\� @/K� b(aa′) = e��� b�� �>rF� �Ù¼�Ðaa′ = e(aa′) = (baa′)aa′ = ba(a′a)a′ = ba(ea′) = b(aa′) = e
s���. ¢ ae = a(a′a) = (aa′)a = as�Ù¼�Ð e�Ér �½Ó1px"é¶s���. ����"f a′�Ér a_� %i�"é¶s���.
&ñú~½Ó&ñd�� a+ x = b\�¦ ÉÒ��H ��A�_� õ�&ñ�Ér ç�H_� /BNo�\�¦ #Qb�G>� ��6 x ���Ht� �Ð#�ï�r��.
(1) ���$� Z��H »!lr\� �'aK� 0�¦ �½Ó1px"é¶Ü¼�Ð ��¦ a_� %i�"é¶s� −a��� ç�Hs���.
(2) ï�rd��\� −a\�¦ �8 �#� (−a) + (a+ x) = (−a) + b.
(3) ���½+ËZO�gË:�¦ ��6 x ���� ((−a) + a) + x = (−a) + b.
(4) −a�� a_� %i�"é¶s�Ù¼�Ð (−a) + a = 0e���¦ s�6 x ���� 0 + x = (−a) + b.
(5) 0��H �½Ó1px"é¶s�Ù¼�Ð x = (−a) + b.
s�\�¦ Ä»o�ú~½Ó&ñd�� ax = b (a 6= 0)�¦ ÉÒ��H �õ� q��§K��Ð��.
(1) ���$� Q∗ = Q− {0}��H Y�L!lr\� �'aK� 1�¦ �½Ó1px"é¶Ü¼�Ð ��¦ a_� %i�"é¶s� a−1��� ç�Hs���.
(2) ï�rd��\� a−1\�¦ Y�L �#� a−1(ax) = a−1b.
(3) ���½+ËZO�gË:�¦ ��6 x ���� (a−1a)x = a−1b.
(4) a−1�� a_� %i�"é¶s�Ù¼�Ð a−1a = 1e���¦ s�6 x ���� 1x = a−1b.
(5) 1�Ér �½Ó1px"é¶s�Ù¼�Ð x = a−1b.
s�%�!3� ¿º ~½Ó&ñd���¦ ÉÒ��H õ�&ñ�Ér ÅÒ#Q��� ½�&h���� ���íß�õ���H �'a>�\O�s� Õª ~½ÓZO�s� 1lx{9� ���. ç�Hõ� ~½Ó&ñd��õ�_� �'a>���H ��6£§_� &ño��Ð ����'a�)a��.�� ���¿Ça�h� 1.1.20 G\�¦ìøÍç�Hs��� ���.Õª�Q��� G��ç�Hs� |c�9�¹Ø�æì�r�|��Ére��_�_� a, b ∈G\� @/K� ~½Ó&ñd�� ax = bü< ya = b�� �½Ó�©� K�\�¦ °ú���H �s���.
7£x"î. G��ç�Hs����0A~½Ó&ñd��s� (Ä»{9�ô�Ç)K�\�¦°ú�6£§�Érì�r"î ���.%i��¦�Ðs�l�0AK� G_�ô�Ç "é¶�è a\�¦ �¦&ñ ���. Õª�Q��� e = ea ∈ G�� �>rF�K�"f ea = a�� �)a��. s�]j b ∈ G\�¦ e��_�_� "é¶�è�� ���. Õª�Q��� x = xa,b ∈ G�� �>rF�K�"f ax = b�� $íwn� ��¦
eb = eax = ax = b
8 ]j 1 �©� ç�H
s� $íwn� �Ù¼�Ð e��H ýa8£¤�½Ó1px"é¶s���. ¢ e��_�_� a ∈ G���� a′s� �>rF�K�"f a′a = e�� ÷&Ù¼�Ð y�� "é¶�è��H ýa8£¤%i�"é¶�¦ °ú���H��. ����"f �è&ño� 1.1.19\� _�K� G��H ç�Hs���.�� ���¿Ça�h� 1.1.21 (S, ∗)\�¦ ìøÍç�Hs��� ���. S∗\�¦ S_� éß�"é¶[þt_� |9�½+Ës��� ���� (S∗, ∗)��Hç�Hs� �)a��. S∗\�¦ S_� í5�xjS!B�s��� ô�Ç��.�� ���×e� 1.1.22 (1) ìøÍç�H (Z,×)_� éß�"é¶ç�H�Ér {±1}.
(2) (Q,×)_� éß�"é¶ç�H�Ér Q∗.
(3) (Mn×n(R),×)_� éß�"é¶ç�H�Ér GLn(R)s���.
ç�H G_� "é¶�è_� >hú\�¦ G_� �DÊÁ(order)�� � 9 |G|�Ð ³ðr�ô�Ç��. |G|�� Ä»ô�Çs����Ä»ô�Çç�Hs��� ��¦, ��m���� Áºô�Çç�Hs��� ô�Ç��. 0Aú�� 1��� ç�H�¦ ��ÃZ�ø5� !B�s��� ô�Ç��.�� ��K±ÓÞDº 1.1.23 G\�¦ ç�Hs��� ��¦ a ∈ Gs��� ���. ëß���� an = e�� ÷&��H �����ú ns� �>rF� ���� s��Qô�Ç n×�æ\�"f ���©� ����Ér n�¦ a_� �DÊÁ(order)�� ��¦ |a|�Ð �����·p��. ëß����an = e�� ÷&��H �����ú ns� �>rF� �t� ·ú§Ü¼��� |a| = ∞�� ô�Ç��.
e��_�_� ç�H G\�"f �½Ó1px"é¶_� 0Aú��H 1s� 9, %i�ܼ�Ð 0Aú�� 1��� "é¶�è��H �½Ó1px"é¶÷�rs���. ¢, am = e{9� �9�¹Ø�æì�r�|��Ér (a−1)m = a−m = e��� �s�Ù¼�Ð
|a| = |a−1|
s���.�� ���×e� 1.1.24 (1) »!lrç�H Z\�"f |0| = 1s��¦ %òs������ "é¶�è��H 0Aú�� �¿º Áºô�Ç@/s���.
(2) 1 ∈ Zn_� 0Aú��H ns���.
(3) Z6_� �âĺ, |0| = 1, |1| = 6, |2| = 3, |3| = 2, |4| = 3, |5| = 6s���.�� ���¿Ça�h� 1.1.25 |a| = n <∞�� ���.
(1) am = am (mod n).
(2) am = es���� n | m.
7£x"î. m = qn + r, 0 ≤ r < n�� æ¼��. Õª�Q��� am = (an)qar = ear = ar. ¢, am = es���� r < n = |a|s�Ù¼�Ð r = 0s�#Q"f n | ms���.
Õª�QÙ¼�Ð�� ���¿Ça�h� 1.1.26 |a| = n{9� �9�¹Ø�æì�r�|��Ér ��6£§ ¿º �|�s� $íwn� ���H �s���.
]j 1.2 ]X� ç�H_� &ñ_� 9
(1) an = e.
(2) ak = es���� n | k.�� ���×e� 1.1.27 (a, b) ∈ G1 ×G2s��¦ |a| = m, |b| = ns���� |(a, b)| = lcm(m,n)s���.
7£x"î. l = lcm(m,n)s����¦ ���� l = mm1, l = nn1õ� °ú s� jþt ú e����. Õª�Q���(a, b)l = ((am)m1 , (bn)n1) = (e, e)
s���. ¢ (a, b)k = (e, e)s���� ak = e, bk = es�Ù¼�Ð m | k, n | ks�#�"f l | k�� �)a��. &ño�1.1.26\� _�K� |(a, b)| = ls���.�� ���¿Ça�h� 1.1.28 ëß���� |a| = n <∞s���� |ak| = n
(n, k)s���.
7£x"î. n = n′(n, k), k = k′(n, k)�� æ¼���(ak)n′ = ak′n′(n,k) = (an)k′ = e
s���.ëß���� (ak)m = es���� akm = es�Ù¼�Ð n | km,7£¤ n′ | k′ms���.Õª���X< (n′, k′) = 1s�Ù¼�Ð n′ | ms���. &ño� 1.1.26\� _�K� |ak| = n′s���.�� ���×e� 1.1.29 Z12_� �âĺ l� ñ�Ð 1ks� k1 = ke���¦ ÅÒ3lq ��¦ |k| = 12/(12, k)�ÐÂÒ'� ��6£§�¦ ~1�>� %3���H��.
k 0 1 2 3 4 5 6 7 8 9 10 11
|k| 1 12 6 4 3 12 2 12 3 4 6 12�� ���×e� 1.1.30 g ∈ GLm(C)_� 0Aú�� ns��� ���� gn = Ims�Ù¼�Ð g_� þj�è���½Ód���Érxn − 1�¦ ��è�H��. Õª���X< xn − 1_� ��H�Ér e2πi/k (k = 1, 2, . . . , n − 1)ܼ�Ð "f�Ð ��ØÔ��.
Õª�QÙ¼�Ð g_� þj�è���½Ód��� "f�Ð ���Ér ��H�¦ °ú�>� �)a��. ����"f g��H @/y���o�� ��0px � 9�¦Ä»u���H xn − 1_� ��Hs�#��� �Ù¼�Ð &h�{©�ô�Ç '��§>= P�� �>rF�K�"f
g = P
(λ1 0 ··· 00 λ2 ··· 0··· ··· ··· ···0 0 ··· λm
)P−1, λn
i = 1
s���. ժ�Q���gk = P
(λk1 0 ··· 0
0 λk2 ··· 0
··· ··· ··· ···0 0 ··· λk
m
)P−1
s�Ù¼�Ð |g| = n{9� �9�¹Ø�æì�r�|��Ér e��_�_� k < n\� @/K� λki 6= 1��� λi�� �>rF� ���H �s�
��.
10 ]j 1 �©� ç�H
V� 1.2 â� )K��â ÌfCÁþ�ÊÁ�� ��K±ÓÞDº 1.2.1 (G, ∗), (G′,#)\�¦ ç�Hs��� ���. ���H a, b ∈ G\� @/K�f(a ∗ b) = f(a)#f(b) (1.4)
\�¦ ëß�7ᤠ���H �<Êú f : G → G′�¦ )K��â ÌfCÁþ�ÊÁ(homomorphism)s��� � 9 ���éß������ ï�r1lx+þA�<Êú\�¦ �â ÌfCÁþ�ÊÁ(isomorphism)s��� ô�Ç��. G = G′��� �âĺ\���H ï�r1lx+þA�<Êú\�¦ ��e�)K��â ÌfCÁþ�ÊÁ(endomorphism), 1lx+þA�<Êú\�¦ ��e��â ÌfCÁþ�ÊÁ(automorphism)���¦ ô�Ç��. f :
G → G′��� 1lx+þA�<Êú�� �>rF� ���� Gü< G′��H �â ÌfC(isomorphic)s����¦ � 9 G ' G′ܼ�Ð ³ðr�ô�Ç��. f : G→ G′��� �����ï�r1lx+þAs� �>rF� ���� G′�¦ G_� )K��â ÌfC(�×(homomorphic
image)���¦ ô�Ç��.
f : G → G′, ψ : G′ → G′′s� ç�H��s�_� (ï�r)1lx+þA�<Êús���� ψ ◦ f : G → G′′� (ï�r)1lx+þA�<Êúe���¦ ·ú� ú e����. ¢ f : G → G′s� 1lx+þA�<Êús���� f−1 : G′ → G� 1lx+þA�<Êús���. Õª�QÙ¼�Ð ç�H��s�_� 1lx+þA�'a>���H 1lxu��'a>�s���. ëß���� G ' G′�� �<Êú f\� _�K� 1lx+þA��� �âĺ g ∈ G\�¦ f(g)�Ð s�2£§�¦ ��Ë#Q Òqty�� ���� ¿º ç�H�Ér °ú ��t���H �s���. z�YV�Ð
N = {n(x) = ( 1 x0 1 ) | x ∈ R}
s����¦ ���� N�Ér '��§>=_� Y�L\� _�K� ç�H�¦ s�ÀÒ 9 n(x)n(y) = n(x+ y)s�Ù¼�Ðf : N → R, n(x) 7→ x
��H 1lx+þA�<Êú�� �)a��. s�]j n(x)\�¦ x���¦ s�2£§�¦ ��Ë��� N = Rs� �)a��!
���H ç�H[þt_� |9�½+Ë 0A_� 1lx+þA�'a>�\� _�ô�Ç ì�r½+É�Ð ëß�[þt#Qt���H 1lxu�ÀÓ[þt_� @/³ð\�¦ &ñ ���H ��Ér ç�H�¦ ì�rÀÓ ���H �s�Ù¼�Ð ç�H�:r_� B�ĺ ×�æ¹ô�Ç 3lq³ð×�æ\� ����� �)a��. ĺo���H0Aú�� q��§&h� ����Ér ç�H[þtõ� Ä»ô�ÇÒqt$í��6\�ç�H[þt�¦ ì�rÀÓ ���H ���\O��¦ ½+É �s���.
ç�H G_� ��l�1lx+þA�<Êú ����_� |9�½+Ë AutG�Ér ½+Ë$í���íß�\� _�K� ç�Hs� H�d�¦ ~1�>� �Ð{9�ú e����. AutG��H G_� @/ú&h�½�_� @/g�A$í�¦ 8£¤|¾Ó ���H ç�Hs���.�� ���×e� 1.2.2 ç�H G_� e��_�_� "é¶�è g\� @/K�
ιg : G→ G, x 7→ gxg−1 (1.5)
��H 1lx+þA�<Êús���. s��Qô�Ç 1lx+þA�<Êú\�¦ 6�ÉÙ��e��â ÌfCÁþ�ÊÁ(inner automorphism) ¢��H ärQc�â ÌfCÁþ�ÊÁ(conjugation)s����¦ � 9?/ÂÒ1lx+þA�<Êú����_�|9�½+Ë�¦ InnG����H��. ¢e��_�_� g\� @/K� gxg−1\�¦ x_� ärQc(conjuagte)���¦ � 9 g_� (��YU �¿º_� |9�½+Ë�¦ g_� ärQcÇÚ(conjugate class)���¦ ô�Ç��. ιaιb = ιabs�Ù¼�Ð ι : G→ AutG, a 7→ ιa��H ï�r1lx+þA�<Êús���.
]j 1.2 ]X� ï�r1lx+þA�<Êú 11
�� ���×e� 1.2.3 |X| = ns���� SX ' Sns���. z�]j�Ð λ : X → {1, 2, . . . , n}�¦ ���éß����<Êú���¦ ���� �<Êú SX → Sn, ϕ 7→ λ−1 ◦ ϕ ◦ λ�Ér 1lx+þA�<Êús���.
1lx+þA��� ç�H[þtz�o���H �¿º /BNÄ»K���ëß� ���H $í|9��¦ Ä©�¿�\� Å]���s����¦ � 9 s� ×�æ:£¤y� s��½Ó���íß�\� _��>r �#� ³ð�&³÷&��H $í|9�[þt�¦ 7�ÊÁ�\� Å]���s����¦ ô�Ç��. \V\�¦ [þt��� ç�HG\� @/K�
(1) |G| = ns���.
(2) g5 = e��� "é¶�è g�� 10>h e����.
(3) e��_�_� a, b ∈ G\� @/K� ab = bas���.
(4) e��_�_� a ∈ G\� @/K� x2 = a��� x ∈ G�� �>rF�ô�Ç��.
1px�Ér ½�&h� $í|9�s� 9 s� ×�æ (2),(3),(4)��H @/ú&h� $í|9�s���. Õª�Q�� ‘2 ∈ Gs���’, ‘G_�"é¶�è��H �<Êús���’ 1px�Ér ½�&h� $í|9�s� ��m���. ¿º ç�Hs� 1lx+þAe���¦ �Ðs��9��� 1lx+þA�<Êú\�¦¹1Ôt�ëß� ¿º ç�Hs� 1lx+þAs� ��_���¦ �Ðs��9��� �Ð:�x ¿º ç�Hs� /BNÄ» �t� 3lw ���H ½�&h� $í|9��¦¹1Ô��H��.�� ���×e� 1.2.4 �<Êú x 7→ ex\� _�K� (R,+) ' (R+,×)s���.�� ���×e� 1.2.5 (R∗,×)ü< (C∗,×)��H1lx+þAs���m���.z�]j�Ð C∗\�"f��He��_�_� a ∈ C∗\�@/K� ~½Ó&ñd�� x2 = a�Ér K�\�¦ °ú�ܼ�� R∗\�"f��H \V�Ð x2 = −1s� K�\�¦ °ú�t� ·ú§Ü¼Ù¼�Ð s� "î]j�� $íwn� �t� ·ú§l� M:ë�Hs���.�� ���×e� 1.2.6 ��6£§�Ér �¿º ï�r1lx+þA�<Êú_� \Vs���.
• e��_�_� &ñú n\� @/K� Z → Z, a 7→ na.
• GLn(R) → R∗, g 7→ det g.
• Z → Zn, a 7→ a.
• e��_�_� z�ú r\� @/K� C(R) → R, ϕ 7→ ϕ(r).
• C(R) → R, ϕ 7→∫ ba ϕdx.
• G1 ×G2 × · · · ×Gn → Gi, (a1, a2, . . . , an) 7→ ai.
• ¿º 7�'�/BNçß���s�_� {9�����8�(linear transformation).�� ���¿Ça�h� 1.2.7 f : G→ G′\�¦ ï�r1lx+þA�<Êú�� ����
12 ]j 1 �©� ç�H
(1) f(e) = e.
(2) e��_�_� &ñú n\� @/K� f(gn) = f(g)ns���. :£¤y� |g| < ∞s���� |f(g)|��H |g|_� ���ús� 9, f�� 1lx+þAs���� |g| = |f(g)|.
7£¤, ï�r1lx+þA�<Êú��H ç�H_� ���íß�, �½Ó1px"é¶, %i�"é¶�¦ �¿º Ä»t�r�&�ï�r��.
Õª�QÙ¼�Ð �����ï�r1lx+þA f : G→ G′�Ér G_� ç�Hܼ�Ð"f_� $í|9��¦ ï�r1lx+þA�©� G′\� ÈÒ%ò ���H �ܼ�Ð, G′�¦ ��Ö�¦s����¦ Òqty�� ���� �)a��. f�� éß����� ����� �âĺ��H G′s� G_� $í|9��¦�¿º {���¦ e��t���H ·ú§t�ëß�, �y��9 G�� B�ĺ ß¼�¦ 4�¤ú�ô�Ç �âĺ q��§&h� ����¦ éß�í�Hô�Ç ç�H��� ï�r1lx+þA�©� G′s� �>rF� ���� G_� #Q�"� $í|9��Ér G′_� $í|9��ÐÂÒ'� ~1�>� Ä»ÆÒ|c ú e����.
ï�r1lx+þA�<Êú\� _�K� "é¶A� ç�H_� $í|9��¦ \O����� {9�>� ÷&��H��\�¦ 8£¤|¾Ó ���H �s� ��6£§_� Ùþ�s�����H �s���.�� ��K±ÓÞDº 1.2.8 f : G→ G′s� ï�r1lx+þA�<Êú{9� M:,
Ker f = {g ∈ G | f(g) = e}
\�¦ f_� h9cs����¦ ô�Ç��.�� ���¿Ça�h� 1.2.9 f�� 1-1{9� �9�¹Ø�æì�r�|��Ér Ker f = {e}��� �s���.
7£x"î. f(a) = f(b) ⇐⇒ e = f(a)f(b)−1 = f(a)f(b−1) = f(ab−1) ⇐⇒ ab ∈ Ker f .
�� ���×e� 1.2.10 �Ðl� 1.2.6\�"f_� ï�r1lx+þA�<Êú_� Ùþ��¦ >�íß�ô�Ç��.
• Z → Z, a 7→ na_� Ùþ��Ér {0}.
• GLn(R) → R∗, g 7→ det g_� Ùþ��Ér SLn(R).
• Z → Zn, a 7→ a_� Ùþ��Ér nZ.
• C(R) → R, ϕ 7→ ϕ(r)_� Ùþ��Ér {ϕ ∈ C(R) | ϕ(r) = 0}.
• C(R) → R, ϕ 7→∫ ba ϕ(x) dx_� Ùþ��Ér {ϕ ∈ C(R) |
∫ ba ϕ(x) dx = 0}.
• G1×G2× · · · ×Gn → Gi, (a1, a2, . . . , an) 7→ ai_� Ùþ��Ér {(a1, a2, . . . , an) | ai = e}.
]j 1.3 ]X� ÂÒì�rç�H 13
V� 1.3 â� ÉÙ&P�!B��� ��K±ÓÞDº 1.3.1 ç�H G_� ÂÒì�r|9�½+Ë H�� G_� ���íß�\� _�K� ç�Hs� |c M:, H\�¦ G_� ÉÙ&P�!B�(subgroup)s����¦ � 9 H ≤ G�Ð �����·p��.
e��_�_�ç�H G\�@/K� {e}ü< G������Ér�½Ó�©�ÂÒì�rç�Hs���. {e}�¦��ÃZ�ø5� ÉÙ&P�!B�(trivial
subgroup)s����¦ ��¦ G�� ����� ÂÒì�rç�H�¦ ë�>ÉÙ&P�!B�s��� ô�Ç��.�� ���×e� 1.3.2 e��_�_� &ñú n\� @/K� nZ = {nk | k ∈ Z}�Ér Z_� ÂÒì�rç�Hs��� (R∗,×)��H(R,+)_� ÂÒì�rç�Hs� ��m���.�� ���¿Ça�h� 1.3.3 G_� ÂÒì�r|9�½+Ë H 6= {}\� @/K� ��6£§_� y�� �|��Ér 1lxu�s���.
(1) H�� G_� ÂÒì�rç�Hs���.
(2) ���H a, b ∈ H\� @/K� ab ∈ H, a−1 ∈ H.
(3) ���H a, b ∈ H\� @/K� ab−1 ∈ H.
(4) H�� Ä»ô�Ç|9�½+Ë��� �âĺ��H ���H a, b ∈ H\� @/K� ab ∈ H.
7£x"î. ���$� H ⊂ Gs�Ù¼�Ð ���½+ËZO�gË:�Ér {©����y� $íwn��<Ê�¦ ÅÒ3lq ���. (1) ⇒ (2) ⇒ (3)��H{©���� ���. (3)�¦ ��&ñ ���. Õª�Q��� H\�"f ô�Ç "é¶�è a\�¦ ×þ� �#� (3)�¦ s�6 x ���� aa−1 =
e ∈ He���¦ ·ú� ú e����. ¢ e��_�_� b ∈ H\� @/K� eb−1 = b−1 ∈ Hs��¦ ����"f e��_�_�a, b ∈ H\� @/K� ab = a(b−1)−1 ∈ Hs�Ù¼�Ð H��H ç�Hs���. Õª�QÙ¼�Ð (3) ⇒ (1)s� $íwn�ô�Ç��. s�]j H\�¦ Ä»ô�Ç|9�½+Ës��� ��&ñ ���. (4)\� _�K� e��_�_� a ∈ H\� @/K� aH ⊂ Hs��¦ |aH| = |H|s�Ù¼�Ð aH = Hs���. Õª�QÙ¼�Ð ah = a��� h ∈ H�� �>rF�ô�Ç��. Gîß�\�"fs� d���¦ Òqty�� ��¦ a−1�¦ Y�LK�ÅÒ��� h = a−1ah = a−1a = es�Ù¼�Ð h��H �½Ó1px"é¶s���. ��r�aH = H�ÐÂÒ'� ak = e��� k ∈ H�� �>rF� ���HX< ì�r"îy� k = a−1s���.�� ���×e� 1.3.4 ��6£§�Ér �¿º ÂÒì�rç�H_� \Vs���.
• e��_�_� &ñú n\� @/K�, nZ ≤ Z.
• Z ≤ Q ≤ R ≤ Cs��¦ Q∗ ≤ R∗ ≤ C∗.
• SLn(R) ≤ GLn(R).
• ô�Ç 7�'�/BNçß�_� ���H ÂÒì�r/BNçß�.
• D(R) ≤ C(R).
14 ]j 1 �©� ç�H
• {0, 2, 4, 6} ≤ Z8.�� ���¿Ça�h� 1.3.5 f : G→ G′�� ï�r1lx+þA�<Êú���¦ ����(1) Ker f ≤ Gs���.
(2) Im f ≤ G′s���. {9�ìøÍ&h�ܼ�Ð H ≤ Gs���� f(H) ≤ G′s���.
(3) H ′ ≤ G′s���� Ker f ≤ f−1(H ′) ≤ Gs���.
7£x"î. (1) a, b ∈ Ker fs���� f(a) = f(b) = es�Ù¼�Ðf(ab−1) = f(a)f(b)−1 = e
s� ÷&#Q"f Ker f ≤ Gs���.
(2) a′, b′ ∈ f(H)s���� a, b ∈ H�� �>rF�K�"f f(a) = a′, f(b) = b′s�Ù¼�Ða′(b′)−1 = f(a)f(b)−1 = f(ab−1) ∈ f(H)
�� ÷&#Q"f f(H) ≤ G′.
(3) a, b ∈ f−1(H ′)s���� f(a), f(b) ∈ H ′s�Ù¼�Ðf(ab−1) = f(a)f(b)−1 ∈ H ′
s� ÷&#Q ab−1 ∈ f−1(H ′)s���.
f : G→ G′s� éß���ï�r1lx+þAs���� f : G→ Im f��H 1lx+þA�<Êús���. Im f ≤ G′s�Ù¼�Ð, s��âĺ G\�¦ G′_� ÂÒì�rç�Hܼ�Ð Òqty��½+É Ãº e��ܼ 9 z�]j�Ð ÕªXO�>� ���H ��¦ Gü< Im f\�¦ �â ���ªø5���(identify)���¦ ô�Ç��.�� ���×e� 1.3.6 R∗ → GLn(R), r 7→ diag(r, r, . . . , r)�Éréß���ï�r1lx+þAs�Ù¼�Ð R∗ ≤ GLn(R)�ÐÒqty��½+É Ãº e����.�� ��K±ÓÞDº 1.3.7 ç�H G_� ���H "é¶�èü< ��8���� "é¶�è[þt_� |9�½+Ë Z(G)�¦ G_� ×�æd��s��� ô�Ç��.
7£¤,
Z(G) = {g ∈ G | ∀a ∈ G, ag = ga}�� ���¿Ça�h� 1.3.8 ι : G→ AutG, g 7→ ιg��H Ùþ�s� Z(G)��� ï�r1lx+þA�<Êús���.�� ���¿Ça�h� 1.3.9 {Hα}α∈A�� ç�H G_� ÂÒì�rç�H[þt_� |9�½+Ës���� ∩α∈AHα� G_� ÂÒì�rç�Hs���.
]j 1.3 ]X� ÂÒì�rç�H 15
7£x"î. a, b ∈ ∩α∈AHαs���� e��_�_� α ∈ A\� @/K� a, b ∈ Hαs��¦ ab−1 ∈ Hα�� ÷&Ù¼�Ðab−1 ∈ ∩α∈AHαs���.
Õª�QÙ¼�Ð X�� ç�H G_� e��_�_� ÂÒì�r|9�½+Ës���� X\�¦ �í�<Ê ���H G_� ���H ÂÒì�rç�H[þt_�/BN:�xÂÒì�r�Ér G_� ÂÒì�rç�Hs� ÷& 9 s���Ér X\�¦ �í�<Ê ���H G_� ÂÒì�rç�H ×�æ\�"f ���©� ����Ér�s���. s� ÂÒì�rç�H�¦ 〈X〉�Ð ³ðr� � 9 X\� _�K� �»jÅ]�(generated)�)a G_� ÂÒì�rç�Hs����¦ô�Ç��. s� &ñ_��ÐÂÒ'�
X ⊂ H ≤ Gs���� 〈X〉 ≤ H (1.6)
e���Ér ì�r"î ���. ëß���� G_� Ä»ô�ÇÂÒì�r|9�½+Ë X�� �>rF�K�"f G = 〈X〉�� ÷&��� G\�¦ ËÂø5��»jÅ]�!B�(finitely generated group)s��� ô�Ç��.�� ���¿Ça�h� 1.3.10 X ⊂ G�� ��¦ X−1 = {x−1 | x ∈ X}���¦ ���� 〈X〉��H X ∪X−1_� Ä»ô�Ç>h_� "é¶�è[þt_� Y�L ����_� |9�½+Ës���. 7£¤,
〈X〉 = {xk11 x
k22 · · ·xkm
m | m ∈ N, xi ∈ X, ki ∈ Z}
Õª�QÙ¼�Ð ëß���� e��_�_� x, y ∈ X\� @/K� xy = yxs���� 〈X〉��H ��6\�ç�Hs��¦, �8ç�H����X = {x1, . . . , xn}s����
〈X〉 = {xk11 x
k22 · · ·xkn
n | ki ∈ Z}.
7£x"î. ���$� H = {xk11 x
k22 · · ·xkm
m | m ∈ N, xi ∈ X, ki ∈ Z}���¦ ���� H�� ç�Hs� �)a��.
z�]j�Ð H_� e��_�_� ¿º "é¶�è a = xk11 x
k22 · · ·xkm
m ü< b = yl11 y
l22 · · ·xkn
n \� @/K�ab−1 = xk1
1 xk22 · · ·xkm
m y−lnn y
−kn−1
n−1 · · · y−l11 ∈ H
s���. Õª���X< ëß���� K�� X\�¦ �í�<Ê ���H ÂÒì�rç�Hs���� {©����y� X ∪X−1_� Ä»ô�Ç>h_� "é¶�è[þt_� Y�Lܼ�Ð ³ðr�÷&��H H_� ���H "é¶�è\�¦ �í�<Ê �>� ÷&Ù¼�Ð H�� X\�¦ �í�<Ê ���H ÂÒì�rç�H×�æ\�"f ���©� ����Ér �s���. �� Qt� ÅÒ�©��Ér ~1�>� ·ú� ú e����.�� ���×e� 1.3.11
Q8 = {±1,±i,±j,±k} (1.7)
1 = ( 1 00 1 ) , i =
(i 00 −i
), j =
(0 1−1 0
), k =
(0 ii 0
)∈ GL2(C) (1.8)
s����¦ ���� k = ijs��¦i4 = 1, i2 = j2 = −1, ji = i3j (1.9)
16 ]j 1 �©� ç�H
s� $íwn� �#� Q8�Ér iü< j\� _�K� Òqt$í÷&��H ç�Hs���. y�� "é¶�è_� 0Aú��H|1| = 1, | − 1| = 2, | ± i| = | ± j| = | ± k| = 4
�Ð >�íß��)a��. Q8�¦ ��xjS�¿!B�(quaternion group)s����¦ ô�Ç��.�� ���×e� 1.3.12 Z = 〈1〉��H Ä»ô�ÇÒqt$íç�Hs��� Q��H Ä»ô�ÇÒqt$íç�Hs� ��m���. z�]j�Ð X =
{a1b1, . . . , an
bn}�¦ Q_� e��_�_� Ä»ô�Ç��� ÂÒì�r|9�½+Ës��� ��¦ p�¦ b1, . . . , bn ×�æ #QÖ¼ �� ��
¾ºt� ·ú§��H �èú�� ���. ëß���� 1/p ∈ 〈X〉s���� &h�{©�ô�Ç &ñú k1, . . . , kn\� @/K�1p
=a1k1 + · · ·+ ankn
b1 · · · bns� ÷&#Q�� �m��� p | b1 · · · bns�#Q�� ��� s���H Ô�¦�� ���.�� ���¿Ça�h� 1.3.13 G = 〈X〉s��¦ f, ψ : G → G′�� ï�r1lx+À>�<Êú�Ð"f ���H x ∈ X\� @/K�f(x) = ψ(x)s���� f = ψs���. 7£¤ 〈X〉�ÐÂÒ'�_� ï�r1lx+þA�<Êú��H X\�"f_� �<Êú°ú\� _�K�¢-a���y� ���&ñ�)a��.
7£x"î. &ño� 1.3.10\� _�K� 〈X〉_� e��_�_� "é¶�è��H a = xk11 x
k22 · · ·xkm
m (m ∈ N, xi ∈X, ki ∈ Z) +þAI��Ð �������¦ x ∈ X\� @/K� f(x) = ψ(x)s�Ù¼�Ð
f(a) = f(x1)k1f(x2)k2 · · · f(xm)km = ψ(x1)k1ψ(x2)k2 · · ·ψ(xm)km = ψ(a)
s���.�� ���×e� 1.3.14 f : Z → G�� ï�r1lx+þAs���� f��H f(1)\� _�K� ¢-a���y� ���&ñ�)a��. z�]j�Ðf(n) = f(1)ns���.�� ���×e� 1.3.15 f : Zn → Zms� ï�r1lx+þAs���� f��H f(1)\� _�K� ¢-a���y� ���&ñ÷&l���H ���f(1)\�¦ ��6£§@/�Ð &ñ½+É Ãº��H \O���. z�]j�Ð f(1) = a���¦ ���� e��_�_� &ñú k\� @/K�
f(k) = ka = ka
����H �s��¦, :£¤y�f(0) = f(n) = na
s���. Õª���X< f(0) = 0s�#Q�� �Ù¼�Ð na ≡ 0 (mod m)s���. Õª�QÙ¼�Ða ≡ 0 (mod m/(n,m))
s�#Q�� ô�Ç��. ìøÍ@/�Ð a�� s� �|��¦ ëß�7ᤠ���� f(k) = ka�� ú� &ñ_�÷&��H ï�r1lx+þA�<Êúe���¦ ~1�>� �Ð{9� ú e����. Õª�QÙ¼�Ð f : Zn → Zm��� ï�r1lx+þA�<Êú f��H (n.m)>h e����.
]j 1.3 ]X� ÂÒì�rç�H 17
X = {g}s���� 〈X〉 = 〈g〉\�¦ g\� _�K� Òqt$í�)a 'K�».É!B�(cyclic group)s��� � 9 g\�¦〈g〉_� �»jÅ]���(generator)���¦ ô�Ç��. &ño� 1.3.10ܼ�ÐÂÒ'�
〈g〉 = {gn | n ∈ Z}
e���¦ ·ú� ú e���¦ |〈g〉| = |g|s���.
ç�H G_� e��_�_� ¿º ÂÒì�r|9�½+Ë A, B\� @/K�,
AB = {ab | a ∈ A, b ∈ B} (1.10)
\�¦ Aü< B_� Y�Ls��� ô�Ç��. A = {a}��� �âĺ AB = aB, B = {b}��� �âĺ AB = Ab�Ðçß�éß�y� ��H��. e��_�_� [j ÂÒì�r|9�½+Ë A,B,C\� @/K� (AB)C = A(BC)s� $íwn�ô�Ç��. Óüt�:rG�� ��6\�ç�H��� �âĺ AB��H A+B�Ð aB = a+B 1pxܼ�Ð �����·p��.�� ���¿Ça�h� 1.3.16 H,K�� G_� ÂÒì�rç�H{9� M:, HK�� G_� ÂÒì�rç�Hs� |c �9�¹Ø�æì�r�|��ÉrHK = KH�� $íwn� ���H �s���. Õª�QÙ¼�Ð G�� ��6\�ç�Hs���� HK��H �½Ó�©� ÂÒì�rç�Hs� �)a��.
7£x"î. HK ≤ Gs���� e��_�_� h ∈ H, k ∈ K\� @/K� kh = (h−1k−1)−1 ∈ HKs�Ù¼�ÐKH ⊂ HKs��¦ ��ðøÍ��t��Ð HK ⊂ KHs���. %i�ܼ�Ð KH = KHs����
(HK)(HK) = H(KH)K = H(HK)K = (HH)(KK) = HK
s�Ù¼�Ð HK��H Y�L!lr\� �'aK� {��)�e���¦ e��_�_� h ∈ H, k ∈ K\� @/K�(hk)−1 = k−1h−1 ∈ KH = HK
s�Ù¼�Ð %i�"é¶\� �'aK�"f� {��)�e��ܼټ�Ð HK��H ÂÒì�rç�Hs���.
18 ]j 1 �©� ç�H
V� 2 *�×
'K�».É!B�, n�».É!B�, �ßjÝ~!B�
19
20 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H
V� 2.1 â� 'K�».É!B��� ��Ça�h� 2.1.1 (1) í�H8�ç�H_� ÂÒì�rç�H�Ér í�H8�ç�Hs���.
(2) Áºô�Çí�H8�ç�H�Ér Zü< 1lx+þAs��¦, 0Aú�� n��� í�H8�ç�H�Ér Znõ� 1lx+þAs���. 7£¤ 0Aú�� °ú �Ér ¿º í�H8�ç�H�Ér 1lx+þAs���.
7£x"î. (1) G = 〈g〉 6= {e}\�¦ í�H8�ç�Hs��� ��¦ H 6= {e}\�¦ G_� ÂÒì�rç�Hs��� ���.
gk ∈ Hs���� g−k ∈ Hs�Ù¼�Ð gk ∈ H�� ÷&��H �����ú k�� �>rF� ��¦ s� ×�æ ���©�����Ér �����ú\�¦ ms��� ���. Õª�Q��� H = 〈gm〉s� �)a��. z�]j�Ð gm ∈ Hs�Ù¼�Ð〈gm〉 ⊂ He���Ér ��"î ��¦, %i�ܼ�Ð gk ∈ Hs���� k = qm+ r (0 ≤ r < n) s����¦ +�"f gr = gk(gm)−q ∈ H�Ð ÂÒ'� r = 0e���¦ %3�ܼټ�Ð gk = (gm)q ∈ Hs���.
(2) G = 〈g〉�� Áºô�Çí�H8�ç�Hs���� �<ÊúZ → G, n 7→ gn
s� 1lx+þAs��¦ G = 〈g〉�� 0Aú�� n��� í�H8�ç�Hs���� �<ÊúZn → G, k 7→ gk
�� ú� &ñ_�÷&��H 1lx+þA�<Êúe���¦ ~1�>� �Ð{9� ú e����.
�� ���¿Ça�h� 2.1.1 (1) Áºô�Çí�H8�ç�H G = 〈g〉_� ÂÒì�rç�H�Ér 6£§s� ����� &ñú k\� @/K� 〈gk〉g1Js���.
(2) G = 〈g〉�� 0Aú�� n > 0��� í�H8�ç�Hs���� e��_�_� n_� ���ú d\� @/K� 0Aú�� d��� ÂÒì�rç�Hs� Ä»{9� �>� �>rF� � 9 Õª��Ér 〈gn/d〉s���.
7£x"î. G = 〈g〉\�¦ í�H8�ç�Hs��� ��¦ H 6= {e}\�¦ ÂÒì�rç�Hs��� ���. 0A &ño�_� 7£x"îܼ�ÐÂÒ'� gm ∈ H��� ���©� ����Ér �����ú m\� @/K� H = 〈gm〉s���. Õª�QÙ¼�Ð (1)�Ér $íwn�ô�Ç��. s�]j |G| = n < ∞s��¦ |H| = d���¦ ���. |gm| = ds�Ù¼�Ð &ño� 1.1.28�ÐÂÒ'� d = n/(n,m)s���. m = (n,m)m′s����¦ æ¼��� gm = (gn/d)m′ ∈ 〈gn/d〉s�Ù¼�ÐH ⊂ 〈gn/d〉s��¦ |〈gn/d〉| = d = |H|�� ì�r"î �Ù¼�Ð H = 〈gn/d〉s���.�� ���¿Ça�h� 2.1.2 (1) Z_� Òqt$í"é¶�Ér 1,−1÷�rs���.
(2) a ∈ Zn�� Zn_� Òqt$í"é¶s� |c �9�¹Ø�æì�r�|��Ér (a, n) = 1��� �s���. Õª�QÙ¼�Ð Zn_�Òqt$í"é¶_� >hú��H φ(n) = |Z∗
n|s���.
7£x"î. (1)�Ér {©���� � 9 (2)��H |a| = n/(n, a)ܼ�ÐÂÒ'� {©���� ���.
]j 2.1 ]X� í�H8�ç�H 21
�� ���¿Ça�h� 2.1.3 H = 〈a〉, K = 〈b〉 �� 0Aú�� y��y�� m,n��� í�H�rç�Hs����, H ×K�� í�H8�ç�H{9� �9�¹Ø�æì�r�|��Ér (m,n) = 1��� �s���. s� �âĺ, H ×K��H (a, b)\� _�K� Òqt$í÷&��Hí�H8�ç�Hs� �)a��.
7£x"î. ���$� (m,n) = 1s��� ���. Õª�Q��� |(a, b)| = mns���. z�]j�Ð(a, b)mn = ((am)n, (bn)m) = (e, e)
s��¦ ëß���� (a, b)k = (ak, bk) = (e, e)s���� ak = e, bk = e�ÐÂÒ'� m | k, n | k\�¦ %3���HX<(m,n) = 1s�Ù¼�Ð nm | ks���. %i�ܼ�Ð (m,n) = d > 1s���� e��_�_� (g, h) ∈ H ×K\� @/K�
(g, h)mn/d = ((gm)n/d, (hn)m/d) = (e, e)
s�Ù¼�Ð |(g, h)| < mns� ÷&#Q H ×K��H í�H8�ç�Hs� ��_���¦ ·ú� ú e����.�� ��Ça�h� 2.1.2 (Ça�ÊÁÒeµ) (1) (m,n) = 1s���� Z∗nm ' Z∗
n × Z∗ms���.
(2) n = 2, 4, pk ¢��H 2pk{9� M: ¢ s� M:\�ëß� Z∗n�� í�H8�ç�Hs��� (éß�, p��H 2�� ����� �è
ú). Õª�QÙ¼�Ð Z∗2 = {1}, Z∗
4 ' Z2s��¦Zpk ' Z2pk ' Zpk−pk−1 .
(3) n ≥ 3\� @/K� Z∗2n ' Z2 × Z2n−2s���.
7£x"î. (1)�Ér ×�æ²DG���_� �� Qt� &ño��ÐÂÒ'� ����� 9, (2)��H "é¶r���H_� �>rF�$í\� �'aô�Ç &ño�s���. &ñú�:r �§F�\�¦ �ÃÐ� ���.�� ���×e� 2.1.4 Z∗
100 = Z∗4 × Z∗
25 ' Z2 × Z20.�� ���¿Ça�h� 2.1.5 n�¦ �ª�_� &ñú�� ���� Aut Zn ' Z∗ns���. z�]j�Ð e��_�_� r ∈ Z∗
n\� @/K�µr : Zn → Zn, µr(a) = ra
���¦ &ñ_� ���� �<Êú r 7→ µrs� 0A ¿º ç�H ��s�_� 1lx+þA�<Êú�� �)a��.
7£x"î. ���$� e��_�_� r ∈ Z∗n\� @/K� µr ∈ Aut Zne���¦ �Ðs���.
µr(a+ b) = r(a+ b) = ra+ rb = µr(a) + µr(b)
s�Ù¼�Ð µr�Ér ï�r1lx+þAs���. ¢ µr(a) = µr(b)s���� ra ≡ rb (mod n)���X< (r, n) = 1s�Ù¼�Ða = bs�Ù¼�Ð µr�Ér éß���s��¦ ����"f �����s��� (Zns� Ä»ô�Ç|9�½+Ës�Ù¼�Ð). ����"f µr�Ér �
22 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H
¿º 1lx+þA�<Êús��¦ |AutG| ≥ φ(n)s���. ô�Ǽ#� f ∈ Aut Zn�Ér &ño� 1.3.13\� _�K� f(1)_�°úܼ�Ð ¢-a���y� ���&ñ�)a��. Õª���X< Zn = Im f = 〈f(1)〉s�#Q�� �Ù¼�Ð f(1)� Zn_� Òqt$í"é¶ ÷&#Q�� �Ù¼�Ð f(1) ∈ Z∗
ns��¦ |Aut Zn| ≤ φ(n)s���. s�]jΦ : Aut Zn → Z∗
n, f 7→ f(1)
���¦ &ñ_� ���� Φ��H ú� &ñ_��)a �<Êús���. Õª���X< e��_�_� f, g ∈ Aut Zn\� @/K�Φ(f ◦ g) = f(g(1)) = g(1)f(1)
s�Ù¼�Ð Φ��H ï�r1lx+þAs� 9 Φ(f) = Φ(g)s���� f(1) = g(1)s�Ù¼�Ð f = gs�#Q"f Φ��H 1-1s���. ����"f Φ��H ������� ÷&#Q 1lx+þA�<Êús���. ��t�}��ܼ�Ð e��_�_� f ∈ Aut Zn\� @/K�f = µf(1)s�Ù¼�Ð s� &ño���H 7£x"î÷&%3���.
V� 2.2 â� n�».É!B��� ��Ça�h� 2.2.1 (Cayley�+ Ça�h�) |G| = ns���� G��H Sn_� ÂÒì�rç�Hõ� 1lx+þAs���.
7£x"î. a ∈ G\� @/K�λa : G→ G, λa(x) = ax
�� &ñ_� ���� λa ∈ SGs� 9λa = λb ⇐⇒ a = b
e���¦ ~1�>� ·ú� ú e����. Õª�QÙ¼�Ðλ : G→ SG, a 7→ λa
��H éß���ï�r1lx+þAs���. SG ' Sns�Ù¼�Ð s� &ño���H 7£x"î÷&%3���.
u�8�ç�H Sn_� "é¶�è\�¦ n�».És����¦ � 9 σ ∈ Sn�Ér �Ð:�x(1 2 · · · n
σ(1) σ(2) · · · σ(n)
)
ü< °ú s� ³ðr�ô�Ç��. 0AAᤠ'���Ér =�G í�H"f@/�Ð æ¼t� ·ú§��� a%~��. Õª�QÙ¼�Ð σ−1��H(σ(1) σ(2) · · · σ(n)
1 2 · · · n
)
]j 2.2 ]X� u�8�ç�H 23
s��� jþt ú e����.
σ ∈ Sns��� ���. ëß���� {1, 2, . . . , n}_� ÂÒì�r|9�½+Ë {k1, k2, . . . , kr}s� �>rF�K�"f
σ(k) =
ki+1, x = ki, (1 ≤ i < r)
k1, x = kr
k, k 6= ki
�� $íwn� ���� σ\�¦ U�s��� r��� 'K�».Én�».É(r-cycle), ¢��H r-í�H8�u�8�s����¦ � 9σ = (k1, k2, . . . , kr)
�Ð�����·p��. ¢U�s��� 2���í�H8�u�8��¦¡õ».É(transposition)s����¦ô�Ç��. 1-í�H8�u�8��Ér�¿º �½Ó1px"é¶s��¦ r-í�H8�u�8�_� 0Aú��H re��s� ì�r"î ���.�� ���×e� 2.2.1 S3��H ��6£§_� 6>h_� "é¶�è\�¦ °ú��¦ e����:
e =
(1 2 3
1 2 3
), σ1 =
(1 2 3
2 3 1
), σ2 =
(1 2 3
3 1 2
),
τ1 =
(1 2 3
2 1 3
), τ2 =
(1 2 3
3 2 1
), τ3 =
(1 2 3
2 1 3
)
s��[þt�Ér �¿º í�H�ru�8�ܼ�Ð ����èq ú e����.
e = (1), σ1 = (1, 2, 3), σ2 = (1, 3, 2), τ1 = (2, 3), τ2 = (3, 1), τ3 = (1, 2)
s���. σ = σ1, τ = τ1s��¦ æ¼��� σ2 = σ2, e = σ3, τ2 = στ = τσs�Ù¼�ÐS3 = {e, σ, σ2, τ, στ, σ2τ} (2.1)
�Ð jþt ú e����. s� "é¶�è[þt ��s�_� Y�L�Érσ3 = τ2 = e, τσ = σ2τ (2.2)
e���¦s�6 x ����~1�>�>�íß�½+Éúe����.d�� (2.2)\�¦ S3\�¦Ça��+ �ÐM� ®�N�(defining relation)���¦ ô�Ç��.
σ, τ ∈ Sn{9� M:, ëß���� ���H 1 ≤ k ≤ n\� @/K� σ(k) = k ¢��H τ(k) = ks���� σü< τ��H"��× �¿���¦ ô�Ç��. ¿º í�H8�u�8� (a1, a2, . . . , ar)õ� (b1, b2, . . . , bs)�� "f�Ð �è{9� �9�¹Ø�æì�r�|��Ér {a1, a2, . . . , ar} ∩ {b1, b2, . . . , bs} = {}��� �e���Ér ì�r"î ���.
24 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H�� ���¿�Ça�h� 2.2.1 σü< τ�� "f�Ð �ès���� στ = τσ�� $íwn�ô�Ç��.
7£x"î. ëß���� σ(i) = i, τ(i) = i s���� στ(i) = i = τσ(i)s���. ëß���� σ(i) = j 6= is���� σ(j) 6= js��¦ τ�� σü< "f�Ð �ès�Ù¼�Ð τ(i) = i, τ(j) = js�#Q�� ô�Ç��. Õª�QÙ¼�Ðστ(i) = σ(i) = j = τ(j) = τσ(i)s���.�� ���¿Ça�h� 2.2.2 ���H u�8� σ ∈ Sn�Ér �©����� "f�Ð �è��� í�H8�u�8�[þt_� Y�Lܼ�Ð ³ð�&³÷& 9s� ì�rK���H í�H8�u�8�s� ��������H í�H"fü< 1-í�H8�í�H8��¦ Áºr� ���� Ä»{9� ���.
7£x"î. ú�<Æ&h�) ±ú�ZO�ܼ�Ð 7£x"î ���. n = 1s���� {©���� ���. s�]j k1 ∈ {1, 2, . . . , n}s��� ���. Õª�Q��� k1, σ(k1), σ2(k1), . . .s� �¿º ��\�¦ ú��H \O�ܼټ�Ð σr(k1) = k1��� ���©� ����Ér�����ú rs� �>rF� ��¦, s� M:
k2 = σ(k1), k3 = σ(k2) = σ2(k1), . . . , kr = σ(kr−1) = σr−1(k1)
s��� ���. ��6£§, X = {1, 2, . . . , n} − {k1, . . . , kr}, c1 := (k1, k2, . . . , kr) s��� ���. ëß���� X = {}s���� σ = c1s���. ëß���� X 6= {}s���� σ′ = σ |X�� ���� σ′�Ér X0A\�"f_� u�8�s�Ù¼�Ð S|X|_� "é¶�è�� �¦ ú e����. ) ±ú�ZO�\� _�K� σ′ = c2c3 . . . cm%�!3� �©����� "f�Ð �è��� í�H�ru�8�[þt_� Y�Lܼ�Ð ³ð�&³÷& 9 Õª�Q��� σ = c1c2 . . . cms���.
Ä»{9�$í�¦ �Ðs�l� 0AK�
σ = c1c2 . . . cm = d1d2 . . . dk
�¦�©�����"f�Ð�è���í�H8�u�8�[þt_�Y�Ls����¦ ���. (ci, dj 6= e). 1 ≤ k ≤ n\�@/K�ëß����k�� ���H ci\� ������t� ·ú§Ü¼��� σ(k) = ks�Ù¼�Ð k��H ���H dj\� ������t� ·ú§��H��. ìøÍ@/�Ð ëß���� k�� #Q�"� ci\� ��������� σ(k) 6= ks�Ù¼�Ð k��H #Q�"� dj\� �������� ô�Ç��. s� �âĺ\���H ���H &ñú r\� @/K� σr(k)�� ci, dj\� �������� �Ù¼�Ð ci = djs���. 0A �Ð�&ño�\�¦ +�"f ciü< dj\�¦ ���ì�r ��¦ ) ±ú�ZO��¦ &h�6 x ���� &ño���H 7£x"î�)a��.�� ���×e� 2.2.3 σ = ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14
5 7 9 14 10 11 12 8 3 13 2 6 4 1 )\�¦ í�H8�u�8�_� Y�Lܼ�Ð ³ð�&³K� �Ð��.
���$�1 7→ 5 7→ 10 7→ 13 7→ 4 7→ 14 7→ 1
\�"f í�H8�u�8� ���ú (1, 5, 10, 13, 4, 14)�¦ %3��¦ ��f�� ������t� ·ú§�Ér ú, \V��X< 2�Ð ÂÒ'�r���� ����
2 7→ 7 7→ 12 7→ 6 7→ 11 7→ 2
]j 2.2 ]X� u�8�ç�H 25
\�"f ���ú (2, 7, 12, 6, 11)�¦ %3���H��. ��f�� ������t� ·ú§�Ér ú, \V��X< 3\�"fÂÒ'�3 7→ 9 7→ 3
s�Ù¼�Ð ���ú (3, 9)\�¦ %3���H��. s�]j ������t� ·ú§�Ér ú��H 8÷�rs�Ù¼�Ð ½ ���H Y�L�Érσ = (1, 5, 10, 13, 4, 14)(2, 7, 12, 6, 11)(3, 9)(8)
s���.�� ������Ça�h� 2.2.4 ���H u�8��Ér ñ8�[þt_� Y�Lܼ�Ð ³ð�&³�)a��.
7£x"î. 0A &ño�\� ���� u�8��Ér í�H8�u�8�[þt_� Y�Ls� 9 y�� í�H8�u�8��Ér(k1, k2, . . . , kr) = (k1, kr)(k1, kr−1) · · · (k1, k2)
ü< °ú s� ñ8�_� Y�Le���¦ s�6 x ���� �)a��.�� ��ÌÁ�+ 2.2.5 ��6£§_� \V\�"f �Ð1pws� u�8��¦ ñ8�[þt_� Y�Lܼ�Ð æ¼��H ~½ÓZO��Ér Ä»{9� �t� ·ú§��
(1, 2, 3, 4, 5) = (4, 5)(3, 5)(2, 5)(1, 5)
= (5, 2)(4, 2)(3, 2)(1, 5)
= (5, 3)(2, 1)(3, 5)(4, 5)(2, 3)(3, 5)
��6£§ d��n1 + n2 + · · ·+ nk = n, n1 ≤ n2 ≤ · · · ≤ nk (2.3)
�¦ ëß�7ᤠ���H �����ú[þt_� í�H"f�©� (n1, n2, . . . , nk)\�¦ n_� &P�TÒ»s����¦ ��¦ n_� "f�Ð ���Ér ì�r½+É_� ú\�¦ n_� &P�TÒ»ÊÁ(partition number)���¦ � 9 p(n)ܼ�Ð ³ðr�ô�Ç��.
p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5
1px�Ér ~1�>� >�íß�½+É Ãº e����. z�]j�Ð n = 4_� ì�r½+É�Ér (1, 1, 1, 1), (1, 1, 2), (1, 3), (2, 2),
(4)s� 5>h�� e��ܼټ�Ð p(4) = 5s���. σ ∈ Sn�¦ 1-í�H8�u�8��¦ �í�<Ê �#� í�H8�u�8�_� U�s��� 7£x�� ��2�¤ �#� "f�Ð �è��� í�H8�u�8�_� Y�Lܼ�Ð +�"f σ = γ1γ2 · · · γks� ÷&%3����¦ ��¦ γi_� U�s�\�¦ ni���¦ ���� (n1, n2, . . . , nk)�Ér n_� ì�r½+És� ÷&��HX< s�\�¦ σ_� 'K�».ÉÄ©�¿(cycle structure)���¦ ô�Ç��. e��_�_� ì�r½+É (n1, n2, . . . , nk)\� @/K�
(1, 2, . . . , n1)(n1 + 1, . . . , n1 + n2) · · · (n1 + · · ·+ nk−1 + 1, . . . , n)
õ� °ú s� í�H8�½��� ÅÒ#Q��� ì�r½+É (n1, n2, . . . , nk)s� ÷&��H u�8�s� �>rF�ô�Ç��.
26 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H�� ���¿Ça�h� 2.2.6 σ ∈ Sn_� í�H8�½��� (n1, n2, . . . , nk)s��� ���.
(1) |σ| = lcm(n1, n2, . . . , nk)s���.
(2) σ_� ���H (��YU_� í�H8�½�� σ_� �õ� °ú �¦, ìøÍ@/�Ð í�H8�½��� σ_� �õ� °ú �Éru�8��Ér σ_� (��YUs���.
7£x"î. (1) σ = γ1γ2 · · · γr(γi[þt�Ér �©����� "f�Ð �è��� ni-í�H8�u�8�)ܼ�Ð æ¼�¦l = lcm(n1, n2, . . . , nk)
s��� Z�~��. |γi| = nis��¦ γi�� �©����� "f�Ð �è��� u�8�s�Ù¼�Ðσl = γl
1γl2 · · · γl
r = e
s���. ëß����σk = γk
1γk2 · · · γk
r = e
s���� γki[þt%i�r��©�����"f�Ð�è���í�H8�u�8�s�Ù¼�Ðs�d���Ér e = σk�¦�©�����"f�Ð
�è��� í�H8�u�8�[þt_� Y�Lܼ�Ð ��H �s���. Õª���X< e = (1)s�Ù¼�Ð &ño� 2.2.2_� Ä»{9�$í\� _�K� ���H i\� @/K� γk
i = es��¦ ����"f ni | ks�Ù¼�Ð l | ks���. &ño� 1.1.26\�_�K� |σ| = ls���.
(2) ���$� e��_�_� τ ∈ Snõ� e��_�_� u�8� γ\� @/K� γ(i) = j�� ����τγτ−1(τ(i)) = τγ(i) = τ(j),
7£¤ τγτ−1 : τ(i) 7→ τ(j)s���. :£¤y�τ(k1, k2, . . . , ks)τ−1 = (τ(k1), τ(k2), . . . , τ(ks)) (2.4)
s�Ù¼�Ðτστ−1 = (τγ1τ
−1)(τγ2τ−1) · · · (τγrτ
−1)
_� í�H8�½���H σ_� �õ� °ú ��. ìøÍ@/�Ð µ ∈ Sn_� í�H8�½��� σ_� �õ� °ú ܼ���σ = (a1, . . . , an1)(an1+1, . . . , an1+n2) · · · (an−nk+1, . . . , an)
µ = (b1, . . . , bn1)(bn1+1, . . . , bn1+n2) · · · (bn−nk+1, . . . , bn)
õ� °ú s� jþt ú e���¦, τ(ai) = bi���¦ &ñ_� ���� d�� 2.4\� _�K� τστ−1 = µs���.
]j 2.3 ]X� u�8�ç�H 27
�� ���×e� 2.2.7 S5_� "é¶�è[þt_� 0Aú\�¦ ½K��Ð��. S5_� "é¶�è_� í�H8�½���H (1, 1, 1, 1, 1),
(1, 1, 1, 2), (1, 1, 3), (1, 4), (1, 2, 2), (2, 3), 5 ×�æ ���s��¦ @/6£x÷&��H 0Aú��H 1, 2, 3, 4, 2,
6, 5s���. \V��X< |(1, 3)(2, 4, 5)| = 6s���.�� ������Ça�h� 2.2.8 Sn_� (��YUÀÓ_� ú��H p(n)s���.�� ���¿Ça�h� 2.2.9 ëß���� σ ∈ Sns� r>h_� ñ8�[þt_� Y�Lܼ�Ð� s>h_� ñ8�[þt_� Y�Lܼ�Ð� ³ð�&³÷&��� r ≡ s (mod 2)s���.
7£x"î. ���½Ód�� P =∏
i<j(xi − xj)\�¦ Òqty�� ��¦ e��_�_� τ ∈ Sn\� @/K�
τ(P ) =∏i<j
(xτ(i) − xτ(j))
�� Z�~��. τ = (k, l) (k < l)s���� P_� ���ú xk−xls� τ(P )\�"f��H xl−xk = −(xk−xl)�Ð���ô�Ç��. ¢ P_� ���ú (xi − xj) ×�æ i, j�� �¿º kü< ���Ér ��Ér τ(P )\�"f Õª@/�Ð ��� �t�·ú§�¦ ���ú�Ð z���� e���¦ �� Qt� P_� ���ú[þt�Ér ±(xi−xk)(xi−xl)�Ð ����¦ t�Ö�¦ ú e����.
#�l�"f ÂÒ ñ��H i, k, l_� ß¼l�\� _�K� &ñK������. Õª���X< τ��H xkü< xl�¦ ��Ë��H �s�Ù¼�Ðs� �©��Ér ����<Ês� \O�s� τ(P )\� ����èß���. Õª�QÙ¼�Ð τ(P ) = −Ps���. ����"f
σ = γ1 · · · γr = δ1 · · · δs
�¦ ñ8�_� Y�Ls��� ���� σ(P ) = (−1)rP = (−1)sP�� ÷&Ù¼�Ð r ≡ s (mod 2)s���.
�� ��K±ÓÞDº 2.2.10 u�8� σ ∈ Sns����ú>h_� ñ8�[þt_�Y�Lܼ�гðr�|cúe��ܼ���ÆÇÐn�».É(even
permutation)s��� ��¦ ÕªXO�t� ·ú§Ü¼��� ©Nên�».É(odd permutation)s����¦ ô�Ç��. ¢ Sn_�ÉÙ¡õÁþ�ÊÁ sgn�Ér ��6£§õ� °ú s� &ñ_��)a��:
sgn(σ) =
1, σ ∈ An
−1, σ 6∈ An
�½Ó1px"é¶�Ér ���u�8�s��¦, ()() = (), ()() = , ()() = , ()() =s� ì�r"î �Ù¼�Ð f.Ëu�8�_� %i�"é¶�Ér f.Ëu�8�s��¦ �<Êú sgn : Sn → {±1}�Ér Ùþ��¦ Anܼ�Ð °ú���H ï�r1lx+þA�<Êúe���Ér ì�r"î ���. Õª�QÙ¼�Ð Sn_� ���u�8� ����_� |9�½+Ë�¦ Ans����¦ ³ðr� ���� An�Ér Sn_� ÂÒì�rç�Hs� ÷&��HX< s�\�¦ ¬7�!B�(alternating group)s��� ô�Ç��.
28 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H
V� 2.3 â� �ßjÝ~!B�e��_�_� @/g�A'��§>= T ∈ GLn(R)\� @/K�
O(T ) = {g ∈ GLn(R) | gTgt = T}
���¦ Z�~��. g ∈ O(T )s���� gTgt = T\�"f T = g−1T (g−1)ts�Ù¼�Ð g−1 ∈ O(T )s���. ¢g, h ∈ O(T )s����
(gh)T (gh)t = g(hTht)gt = gTgt = T
s�Ù¼�Ð gh ∈ O(T )s���. Õª�QÙ¼�Ð O(T ) ≤ GLn(R)s��¦, :£¤y� O(T ) ���� ç�Hs���.
O(T )\�¦ T_� ÒÏ�¬!B�(orthogonal group)s����¦ ÂÒ�Ér��. T = In��� �âĺ O(T ) = O(n)s����¦ æ¼�¦ éß�í�Hy� f���§ç�Hs����¦ � 9 T =
(Ir 00 −Is
)��� �âĺ O(T ) = O(r, s)���¦ ��H��.
g ∈ O(T )s����detT = det gTgt = det g detT det gt = (det g)2 detT
\�"f det g2 = 1s�Ù¼�Ð det(g) = ±1s���. Õª�QÙ¼�Ðdet : O(T ) → {±1}
�Ér �����ï�r1lx+þA�<Êús��¦ Õª_� Ùþ��ÉrSO(T ) = {g ∈ O(T ) | det g = 1}
���X< s���¦ T_� §�ÊÁÒÏ�¬!B�(special orthogonal group)s����¦ ô�Ç��.
U(n) = {g ∈ GLn(C) | ggt = In}
s����¦ ���. g ∈ U(n){9� �9�¹Ø�æì�r�|��Ér g−1 = gt��� �s���. g ∈ U(n)s����(g−1)−1 = g = (gt)−1 = g−1
t
s�Ù¼�Ð g−1 ∈ U(n)s���. ¢ g, h ∈ U(n)s����(gh)−1 = h−1g−1 = htgt = gh
t
s�Ù¼�Ð gh ∈ U(n)s���. ����"f U(n)�Ér GLn(C)_� ÂÒì�rç�Hs���. U(n)�¦ ËÂf���h�!B�s��� ô�Ç��. g ∈ U(n)s���� |det g| = 1s���.
]j 2.3 ]X� '��§>=ç�H 29
¢ J =(
0 In−In 0
)∈ GL2n(R){9� M:,
Spn(R) = {g ∈ GL2n(R) | gJgt = J}
s����¦ Z�~��. 0A\�"fü< ��ðøÍ��t��Ð g ∈ Spn(R)s���� gJgt = J\�"f J = g−1J(g−1)ts�Ù¼�Ð g−1 ∈ Spn(R)s���. ¢ g, h ∈ Spn(R)s����
(gh)J(gh)t = g(hJht)gt = gJgt = J
s�Ù¼�Ð gh ∈ Spn(R)s���. Õª�QÙ¼�Ð Spn(R)�Ér ≤ GL2n(R)_� ÂÒì�rç�Hs���. Spn(R)�¦ ��¬!B�(symplectic group)s����¦ ÂÒ�Ér��.�� ���¿Ça�h� 2.3.1 g =
(a bc d
), a, b, c, d ∈Mn(R)\� @/K� g ∈ Spn(R){9� �9�¹Ø�æì�r�|��Érabt = bat, cdt = dct, adt − bct = In (2.5)
¢��Hcta = atc, dtb = btd, atd− ctb = In (2.6)
��� �s���. s� �âĺ g−1 =(
dt −bt
−ct at
)s���.
7£x"î. d��gJgt =
(−bat+abt −bct+adt
−dat+cbt −dct+cdt
)= J
\�"f d�� (2.5)_� 1lxu��|�s� ���:r��. ô�Ǽ#� g ∈ Spn(R)s���� g−1 ∈ Ss��¦ 1lxu��|�(2.5)�¦ ��6 x ����
h =(
dt −bt
−ct at
)\� @/K� gh = I2ns� H�d�¦ ~1�>� ·ú� ú e��ܼټ�Ð h = g−1s���. s�]j g−1\� 1lxu��|�(2.5)�¦ &h�6 x ���� 1lxu��|� (2.6)s� ���:r��.
n = 1��� �âĺ, 0A_� 1lxu��|��Ér éß�í�Hy� ad − bc = 1s�Ù¼�Ð Sp1(R) = SL2(R)s���.
¢P = {
(a b0 (at)−1
)| a ∈ GLn(R), b = bt ∈Mn(R)}
s��� ���. '��§>=(
a b0 (at)−1
)∈ P��H d�� (2.6)�¦ ëß�7ᤠ�Ù¼�Ð Spn(R)_� "é¶�ès���.(
a b0 (at)−1
)(a1 b10 (at
1)−1
)=(
aa1 ab1+b(at1)−1
0 ((aa1)t)−1
)s��¦ (
a b0 (at)−1
)−1=(
a−1 bt
0 at
)
30 ]j 2 �©� í�H8�ç�H, u�8�ç�H, '��§>=ç�H
s�Ù¼�Ð P�� Spn(R)_� ÂÒì�rç�Hs� H�d�¦ ·ú� ú e����. P\�¦ Siegel ï�ä·ÉÙ&P�!B�s��� ô�Ç��.
e��_�_� 0 ≤ k ≤ n\� @/K�Spk(R)× Spn−k(R) → Spn(R)((
a1 b1c1 d1
),(
a2 b2c2 d2
))7→
(a1 b1
a2 b2c1 d1
c2 d2
)
�Érd�� (2.5)�¦��6 x ����éß���ï�r1lx+þA�<Êúe���¦·ú�úe����.Õª�QÙ¼�Ð Spk(R)×Spn−k(R)��HSpn(R)_� ÂÒì�rç�Hܼ�Ð �¦ ú e����.�� ��Ça�h� 2.3.1 (Bruhat &P�B�) wi =
(Ii 0
0 In−i
0 Ii−In−i 0
)s��� ����
Spn(R) =n∐
i=0
PwiP
s��¦ PwiP = {(
a bc d
)∈ Spn(R) | rank(c) = n− i}s���.�� ���×e� 2.3.2 n = 1s���� P = {
(a b0 a−1
)| a ∈ R∗, b ∈ R}s� 9, w = w0 = J =
(1
−1
),
w1 = I2s�Ù¼�Ð Bruhat ì�rK���HSL2(R) = P ∪ PwP
s���.�� ������Ça�h� 2.3.3 g ∈ Spn(R)s���� det g = 1s���.
7£x"î. detwi = 1s��¦ e��_�_� p ∈ P\� @/K� det p = 1s�Ù¼�Ð Bruhat ì�rK�&ño��ÐÂÒ'�s� &ño���H �����:r��.
V� 3 *�×
�b�#bÇÚÿ? �b�#b!B�
31
32 ]j 3 �©� eç#�ÀÓü< eç#�ç�H
V� 3.1 â� �b�#bÇÚÿ? Lagrange Ça�h�|9�½+Ë X\�"f_� �'a>� ∼⊂ X ×X�� ��6£§ [j��t� �|��¦ ëß�7᤽+É M:, ∼�¦ �â n�®�N����¦ô�Ç��:
(1) e��_�_� x ∈ X\� @/K� x ∼ x (ìøÍ��&h�).
(2) x ∼ ys���� y ∼ x (@/g�A&h�).
(3) x ∼ y, y ∼ zs���� x ∼ z (���s�&h�).
∼s� 1lxu��'a>�{9� M:,
[x] = {y ∈ X | y ∼ x}
�¦ x_� �â n�ÇÚ���¦ � 9 x\�¦ [x]_� 7�ØxjSs����¦ ô�Ç��. e��_�_� x ∈ X\� @/K� x ∈ [x]s� 9, e��_�_� x, y ∈ X\� @/K�
[x] = [y] ¢��H [x] ∩ [y] = ∅
s� $íwn�ô�Ç��. Õª�QÙ¼�Ð X��H "f�Ð ���Ér 1lxu�ÀÓ �¿º_� ½+Ë|9�½+Ës���. 7£¤, y�� 1lxu�ÀÓ[þt�ÐÂÒ'� ���_� @/³ð"é¶m���¦ i(v�� ëß���H @/³ð"é¶[þt_� |9�½+Ë�¦ Rs����¦ ����
X =∐x∈R
[x]
�� $íwn�ô�Ç�� (∐��H "f�Ð �è��� |9�½+Ë[þt_� ½+Ë|9�½+Ëe���¦ ³ðr�ô�Ç��). s�\�¦ ¿º�¦ 1lxu�ÀÓ[þts�
X\�¦ &P�TÒ»ô�Ç���¦ ô�Ç��.
s�]j H\�¦ G_� ÂÒì�rç�Hs��� ���. e��_�_� g ∈ G\� @/K�Hg = {hg | h ∈ H}
\�¦ g\�¦ �í�<Ê ���H G\�"f_� H_� ËÁ¥��b�#bÇÚ,
gH = {gh | h ∈ H}
\�¦ g\�¦�í�<Ê ���H G\�"f_� H_���¥��b�#bÇÚ���¦ � 9 g\�¦ Hgü< gH_�@/³ð"é¶s����¦ô�Ç��. ���H g\� @/K�
H = eH → gH, h 7→ gh
ü<H → Hg, h 7→ hg
]j 3.1 ]X� eç#�ÀÓü< LAGRANGE &ño� 33
��{9�@/{9�@/6£xs�Ù¼�Ðy��eç#�ÀÓ_�"é¶�è_�ú��H �¿º |H|ü<°ú 6£§�¦Ä»_� ���.ýa8£¤eç#�ÀÓ ����_� |9�½+Ë�¦ G/H, ĺ8£¤eç#�ÀÓ ����_� |9�½+Ë�¦ H\G�Ð �����·p��. ��z� eç#�ÀÓ[þt�Ér ��6£§õ� °ú �Ér 1lxu��'a>�\� _�ô�Ç 1lxu�ÀÓ�� �)a��. e��_�_� a, b ∈ G\� @/K�
a ∼ b ⇐⇒ a−1b ∈ H (3.1)
���¦ &ñ_� ���. Õª�Q��� a−1a = e ∈ Hs�Ù¼�Ð a ∼ as� 9 a ∼ bs���� a−1b ∈ Hs�#Q"fb−1a = (a−1b)−1 ∈ Hs�Ù¼�Ð b ∼ a�� $íwn�ô�Ç��. ¢, a ∼ b, b ∼ cs���� a−1b, b−1c ∈ Hs�#Q"f a−1b = (a−1b)(b−1c) ∈ H�� ÷&Ù¼�Ð a ∼ c�� $íwn�ô�Ç��. 7£¤, ∼�Ér 1lxu��'a>�s���. Õª���X<
g ∼ a ⇐⇒ a−1g ∈ H ⇐⇒ g ∈ aH
s�Ù¼�Ð s� 1lxu��'a>�\� _�ô�Ç 1lxu�ÀÓ��H [a] = aH, ýa8£¤eç#�ÀÓs���. 0A\�"f H�� ÂÒì�rç�Hs�����H ��z��¦ ��6 xÙþ¡6£§�¦ ÅÒ3lq ���. ��ðøÍ��t��Ð
a ∼ b ⇐⇒ ab−1 ∈ H (3.2)
���¦ &ñ_�ô�Ç �'a>���H 1lxu��'a>��� ÷&�¦ s� 1lxu��'a>�\� _�ô�Ç 1lxu�ÀÓ��H ĺ8£¤eç#�ÀÓ�� �)a��. 3.1ü< 3.2\� _�K�
aH = bH ⇐⇒ a−1b ∈ Hs��¦ Ha = Hb ⇐⇒ ab−1 ∈ H. (3.3)�� ���×e� 3.1.1 G = S3 = {e, σ, σ2, τ, στ, σ2τ}�� ��� (d�� (2.1)�¦ �Ð��).
H = 〈σ〉 = {e, σ, σ2}
s��� ����ýa8£¤eç#�ÀÓ�Ð��H���$� eH = H��e����HX< σ, σ2 ∈ Hs�Ù¼�Ð σH = σ2H = Hs���. s�]j H\� \O���H "é¶�è�Ð \V��X< τ\�¦ ú�ܼ���
τH = {τ, τσ, τσ2} = {τ, σ2t, στ}
�� ¢ ���_� ýa8£¤eç#�ÀÓ���X< στ, σ2τ ∈ τHs�Ù¼�Ð
στH = σ2τH = τH
s���. Õª�QÙ¼�ÐG/H = {H, τH}
34 ]j 3 �©� eç#�ÀÓü< eç#�ç�H
s���. ĺ8£¤eç#�ÀÓ��H ���Ér ~½ÓZO�ܼ�Ð Òqty��K� �Ð��. H = He�� ���_� ĺ8£¤eç#�ÀÓs��¦|H| = 3 = |G|/2s���. eç#�ÀÓ[þt�Ér °ú �Ér ú(s� �âĺ 3)_� "é¶�è\�¦ °ú�ܼ���"f ���� G\�¦ ì�r½+É �Ù¼�Ð ���Ér eç#�ÀÓ��H ��� ÷�rܼ�Ð G−H�� ÷&#Q�� ô�Ç��. 7£¤
H\G = {H,G−H}
s���.
K = 〈τ〉 = {e, τ}s���� ýa8£¤eç#�ÀÓ��H K = eK�� e���¦, ��6£§Ü¼�Ð K\� \O���H "é¶�è, \V��X< σ\� @/K� σK = {σ, στ} = στK\�¦ %3��¦, ��r� K ∪ σK\� \O���H "é¶�è, \V��X< σ2\�@/K� σ2K = {σ2, σ2τ} = σ2τK\�¦ %3���HX< K ∪ σK ∪ σ2K = Gs�Ù¼�Ð
G/K = {K,σK, σ2K}
s���.��ðøÍ��t��Ðĺ8£¤eç#�ÀÓ\�¦ ½ ���� K, Kσ = {σ, σ2τ} = Kσ2τ , Kσ2 = {σ2, στ} =
Kστü< °ú s� 3>h\�¦ %3���H��. s� �âĺ σK 6= Kσ, σ2K 6= Kσ2e���¦ Ä»_� ���.
H\G→ G/H, Hg 7→ g−1H
�Ð &ñ_�÷&��H �<Êú��H {9�@/{9��<Êús�Ù¼�Рĺ8£¤eç#�ÀÓ_� >húü< ýa8£¤eç#�ÀÓ_� >hú��H °ú ��. s� >hú\�¦ G\�"f_� H_� m�ÊÁ���¦ ��¦ [G : H]�Ð �����·p��. Õª���X< eç#�ÀÓ[þts�G\�¦ ì�r½+É �Ù¼�Ð
|G| = |H|[G : H] (3.4)
s� $íwn�ô�Ç��.�� ���×e� 3.1.2 �§@/ç�H An ≤ Sn�¦ Òqty�� ��¦ f.Ëu�8� τ\�¦ �¦&ñ ���. τ−1� f.Ëu�8�s�Ù¼�Ð e��_�_� f.Ëu�8� σ\� @/K� στ−1��H ���u�8�s��¦ Anστ
−1 = Ans� �)a��. Õª�QÙ¼�Ð σ ∈Anσ = Anτs� ÷&#Q Anτ��H ���H f.Ëu�8��¦ �í�<Êô�Ç��. Õª�QÙ¼�Ð
Sn = An ∪Anτ
s��¦ [Sn : An] = 2�� ÷&#Q d�� (3.4)\� _�K�|An| = |Sn|/[Sn : An] = n!/2
s���.
d�� (3.4)�ÐÂÒ'� ��6£§_� ×�æ¹ô�Ç &ño�\�¦ %3���H��.
]j 3.1 ]X� eç#�ÀÓü< LAGRANGE &ño� 35
�� ��Ça�h� 3.1.1 (Lagrange Ça�h�) ëß���� G�� Ä»ô�Çç�Hs��¦ H�� G_� ÂÒì�rç�Hs���� H_� 0Aú��H G_� 0Aú\�¦ ��è�H��.�� ������Ça�h� 3.1.3 (Euler�+ Ça�h�) n�¦ �ª�_� &ñú�� ���� (n, a) = 1��� e��_�_� &ñú a\�@/K� aφ(n) ≡ 1 (mod n)s���.
7£x"î. (Z∗n,×)�� 0Aú�� φ(n)��� ç�Hs�l� M:ë�Hs���.�� ������Ça�h� 3.1.4 (1) |G| = n < ∞s��¦ g ∈ Gs���� |g|��H n_� ���ús��¦ ����"f gn =
es���.
(2) ëß���� |G|�� �èús���� G��H í�H8�ç�Hs� 9 ����"f ��6\�ç�Hs���.
(3) H,K ≤ Gs��¦ (|H|, |K|) = 1s���� H ∩K = {e}s���.�� ���×e� 3.1.5 0Aú�� 2,3,5,7��� ç�H�Ér y��y�� Z2, Z3, Z5, Z7õ� 1lx+þAs���. G\�¦ 0Aú�� 4���ç�Hs��� ���. ëß���� G�� 0Aú�� 4��� "é¶�è\�¦ �í�<Ê ���� G ' Z4s���. ÕªXO�t� ·ú§Ü¼��� G_����H"é¶�è_�0Aú��H 4_����ús��¦ 4��H��m�Ù¼�Ð 1 ¢��H 2s���.����"f ���H a ∈ G\�@/K� a2 = es��¦ �Ðl� 1.1.7\� _�K� G��H ��6\�ç�Hs���. s�]j a, b\�¦ �½Ó1px"é¶s� ����� G_� ¿º"é¶�è�� ���� ab = ba 6= e, a, bs�Ù¼�Ð G = {e, a, b, ab} = 〈a, b〉s���. �<Êú
Z2 × Z2 → G, (1, 0) 7→ a, (0, 1) 7→ b
s� 1lx+þA�<Êúe���Ér ~1�>� �Ð{9� ú e��ܼټ�Ð G ' Z2 × Z2s���. ����:r&h�ܼ�Ð 0Aú�� 4��� ç�H�Ér Z4 ¢��H Z2 × Z2ü< 1lx+þAs��¦ :£¤y� �¿º ��6\�ç�Hs���.�� ���×e� 3.1.6 ab = bas��¦ (|a|, |b|) = 1s���� |ab| = |a||b|s���.
7£x"î. |a| = m, |b| = ns��� ����(ab)mn = (amn)(bmn) = (am)n(bn)m = ee = e
s���. ëß���� (ab)k = es����ak = b−k ∈ 〈a〉 ∩ 〈b〉 = {e}
s�Ù¼�Ð ak = bk = e. Õª�QÙ¼�Ð m | k, n | ks��¦ (m,n) = 1s�Ù¼�Ð mn | ks���. &ño�1.1.26\� _�K� |ab| = mns���.
�� ���×e� 3.1.7 σ = (1, 2, . . . , n), τ =
(1 2 · · · n
1 n · · · 2
)\�¦ Sn_� ¿º "é¶�è���¦ ��¦
Dn = 〈σ, τ〉
36 ]j 3 �©� eç#�ÀÓü< eç#�ç�H
�� ���. Õª�Q��� |σ| = n, |τ | = 2s� 9 τσ = σ−1τs�l� M:ë�H\� &ño� 1.3.10\� _�K�
Dn = {σiτ j | 0 ≤ i ≤ n− 1, 0 ≤ j ≤ 1} (3.5)
s� ÷&Ù¼�Ð |Dn| ≤ 2ne���¦ ·ú� ú e����. s�]j 〈σ〉��H 0Aú�� n��� ÂÒì�rç�Hs�Ù¼�Ð Dn_� 0Aú��H n_� C�ús���. Õª���X< τ 6= σs�Ù¼�Ð n < |Dn| ≤ 2ns�Ù¼�Ð |Dn| = 2ns� ÷&#Q 0A\�"f��\P�ô�Ç "é¶�è��H �¿º ���Ér "é¶�ès���. Dn�¦ 0Aú�� 2n��� Ça�l�¢�>W�!B�(dihedral group)s����¦ ô�Ç��. τσ = σ−1τ\�"f τστ−1 = σ−1\�¦ %3��¦ s�\�¦ i��[þv]jY�L ����
τσiτ−1 = σ−i
s� ÷&#Q τσi = σ−iτ�� �)a��. ����"f (σiτ)2 = σiτσiτ = σiσ−iττ = e�� ÷&Ù¼�Ð |σiτ | =
2s���. ĺ���y� D3 = S3s��¦ n ≥ 3s���� τσ = σ−1τ 6= στs�Ù¼�Ð Dn�Ér ��6\�ç�Hs� ��_���¦Ä»_� ���.
V� 3.2 â� Ça�ĪÉÙ&P�!B�ø� �b�#b!B��� ��K±ÓÞDº 3.2.1 N�¦ ç�H G_� ÂÒì�rç�Hs��� ���. ëß���� e��_�_� a ∈ G\� @/K� aN = Na�� $íwn� ���� N�¦ G_� Ça�ĪÉÙ&P�!B�s����¦ � 9 N / G���¦ ��H��.
ç�H G_� ��"îô�Ç ÂÒì�rç�H {e}ü< G��H &ñ½©ÂÒì�rç�Hs���. ¢, ëß���� G�� ��6\�ç�Hs���� ���HÂÒì�rç�H�Ér &ñ½©ÂÒì�rç�Hs���. N ≤ H ≤ G{9� M:, N / Gs���� N / Hs��� N / Gs��8���N /H{9� �9�¹��H \O���.�� ���¿Ça�h� 3.2.2 Ns� G_� ÂÒì�rç�H{9� M:, ��6£§�Ér �¿º 1lxu�s���:
(1) N / G.
(2) ���H a ∈ G\� @/K� aNa−1 = N .
(3) ���H a ∈ G\� @/K� aNa−1 ⊂ N .
(4) ���H a, b ∈ G\� @/K� (aN)(bN) = abN .
7£x"î. (1) ⇐⇒ (2)��Hì�r"î ���. (3)s�$íwn� ����e��_�_� a ∈ G\�@/K� a−1Na ⊂ N\�"fN ⊂ aNa−1s�Ù¼�Ð (2)�� $íwn� � 9 (2) ⇒ (3)�Ér ì�r"î ���. N / Gs���� e��_�_� a ∈ G\�@/K� aN = Nas�Ù¼�Ð aNbN = aNNb = aNb = abNs�Ù¼�Ð (1) ⇒ (4)s���. ��t�}��ܼ�Ð (4)�� $íwn� ���� e��_�_� a ∈ G\� @/K� aNa−1 ⊂ aNa−1N ⊂ aa−1NN = Ns�Ù¼�Ð(3)s� $íwn�ô�Ç��.
]j 3.2 ]X� &ñ½©ÂÒì�rç�Hõ� eç#�ç�H 37
N / G�� ���. e��_�_� g ∈ G,n ∈ N\� @/K� n′ = gng−1 ∈ N , n′′ = g−1ng ∈ Ns����¦ ����
gn = n′g, ng = gn′′ (3.6)
s� $íwn�ô�Ç��.�� ��ÌÁ�+ 3.2.3 ëß���� K�� G_�&ñ½©ÂÒì�rç�Hs���m���� (aK)(bK) = abK{9��9�¹��H\O���.z�]j�Ð �Ðl� 3.1.1\�"f σK = {σ, στ}, σ2K = {σ2, σ2τ}s���
σKσK = {σ2, σ2τ, στσ, στστ} = {σ2, σ2τ, τ, e} 6= σ2K
s���.�� ���×e� 3.2.4 [G : N ] = 2s���� N /Gs���.z�]j�Ðeç#�ÀÓ��Héß�¿º>h÷�rs��¦ H��Hýa8£¤eç#�ÀÓs� 9 1lxr�\� ĺ8£¤eç#�ÀÓs�Ù¼�Ð G/N = {N,G−N} = N\Ge���Ér ì�r"î �l� M:ë�Hs���. ½�&h����\V�Ð [Sn, An] = 2s�Ù¼�Ð An /Sns���. ¢ �Ðl� 3.1.7\�"f |〈σ〉| = |σ| = ns��¦ |Dn| = 2ns�Ù¼�Ð 〈σ〉 / Dns���.�� ��ÌÁ�+ 3.2.5 N /K, K / Gs��8��� N / G{9� �9�¹��H \O���. z�]j�Ð
G = D4 = {e, σ, σ2, σ3, τ, στ, σ2τ, σ3τ}
�¦Òqty�� ��¦ (�Ðl� 3.1.7�¦�Ð��) N = {e, τ}, K = {e, σ2, τ, σ2τ}���¦ ú�ܼ��� [K : N ] =
[G : K] = 2s�Ù¼�Ð N /K, K /Gs��� σN = {σ, στ} 6= {σ, τσ}s�Ù¼�Ð N / G�� ��m���.�� ��Ça�h� 3.2.1 N / Gs����G/N = {gN | g ∈ G}
�Ér ���íß� (aN)(bN) = abN\� @/K� ç�Hs� ÷& 9 �<Êú pN : G → G/N , a 7→ aN�Ér Ker f =
N��� ï�r1lx+þA�<Êús���.
7£x"î. ���íß� (aN)(bN) = abNs� ���½+ËZO�gË:�¦ ëß�7á¤�<Ê�Ér ì�r"î ���. e��_�_� aN ∈ G/N\�@/K� (aN)(eN) = aeN = aN = eaN = (eN)(aN)s�Ù¼�Ð eN = Ns��½Ó1px"é¶s���. ¢e��_�_� aN\� @/K� (aN)(a−1N) = (a−1N)(aN) = eN = Ns�Ù¼�Ð aN_� %i�"é¶�Ér a−1Ns���. pNs� ï�r1lx+þA�<Êúe���Ér ì�r"î ���.�� ���×e� 3.2.6 G\�¦ 0Aú�� 6��� ç�Hs����¦ ���. ëß���� G�� 0Aú�� 6��� "é¶�è\�¦ �í�<Ê ����G ' Z6ܼ�Ð &ñK�t�Ù¼�Ð G_� ���H "é¶�è_� 0Aú�� 7£¤ 1,2 ¢��H 3s����¦ ��&ñ ���. ���$� G��H 0Aú�� 3��� "é¶�è\�¦ �í�<Ê ��¦ e��6£§�¦ �Ðs���. ÕªXO�t� ·ú§Ü¼��� e��_�_� a ∈ G\�
38 ]j 3 �©� eç#�ÀÓü< eç#�ç�H
@/K� a2 = e�� ÷&#Q �Ðl� 1.1.7\� _�K� G��H ��6\�ç�Hs� ÷&#Q�� ô�Ç��. s� �âĺ e�� �����"f�Ð ���Ér G_� "é¶�è a, b\�¦ ú�ܼ��� {e, a, b, ab}��H ç�Hs� ÷&#Q G�� 0Aú�� 4��� ÂÒì�rç�H�¦°ú�>� ÷&�¦ s���Ér Lagrange&ño�\� �í�H�)a��. s�]j 0Aú�� 3��� "é¶�è a ∈ G\�¦ ×þ� ��¦H = 〈a〉 = {e, a, a2}s��� Z�~��. [G : H] = 2s�Ù¼�Ð e��_�_� b ∈ G−H\� @/K�
G = H ∪Hb = {e, a, a2, b, ab, a2b} = 〈a, b〉
s��¦ H / Gs���. ����"f G/H��H 0Aú�� 2��� ç�Hs� ÷&�¦ e��_�_� x ∈ Hb\� @/K� Hx =
Hbs�Ù¼�Ð (Hx)2 = (Hb)2 = Hs���. ëß���� x ∈ Hbs��¦ |x| = 3s����Hx = Hx4 = H(x2)2 = ((Hx)2)2 = H2 = H
�� ÷&#Q �í�Hs�Ù¼�Ð Hb_� ���H "é¶�è_� 0Aú��H 2s���. ����"f ba ∈ bH = Hb�ÐÂÒ'�|ba| = 2s�Ù¼�Ð baba = e, 7£¤ bab−1 = a−1 = a2s���. s�]j G = 〈a, b〉, a3 = b2 = e,
ba = a2bs�Ù¼�Ð d�� (2.2)ܼ�ÐÂÒ'� G ' S3 = D3e���¦ ·ú� ú e����. ����:r&h�ܼ�Ð 0Aú��6��� ç�H�Ér Z6 ¢��H S3ü< 1lx+þAs���.�� ��K±ÓÞDº 3.2.7 N/G{9�M:,G/N�¦N\�_�ô�ÇG_��b�#b!B� (factor group) ¢��H(�×!B�(quotient
group)s��� � 9 pN�¦ Ø)K�¦�>��)K��â ÌfC(canonical projection)s����¦ ô�Ç��.
f : G → G′s� ï�r1lx+þA�<Êús���� f(a) = f(b) ⇐⇒ f(a−1b) = f(a)−1f(b) = e ⇐⇒a−1b ∈ Ker f ⇐⇒ aKer f = bKer f, 7£¤
f(a) = f(b) ⇐⇒ aKer f = bKer f (3.7)
s� 9 :£¤y� e��_�_� a ∈ G\� @/K�f−1(f(a)) = {b ∈ G | f(b) = f(a)} = aKer f. (3.8)�� ��Ça�h� 3.2.2 (]j11lx+þA&ño�) f : G→ G′�� ï�r1lx+þAs���� Ker f / Gs��¦
f : G/Ker f ' Im f
aKer f 7→ f(a)
�Ér 1lx+þA�<Êús���.
7£x"î. b ∈ Ker fs���� e��_�_� a ∈ G\� @/K�f(aba−1) = f(a)f(b)f(a)−1 = f(a)f(a)−1 = e
]j 3.2 ]X� &ñ½©ÂÒì�rç�Hõ� eç#�ç�H 39
s�Ù¼�Ð aba−1 ∈ Ker fs�Ù¼�Ð 3.2.2(4)\� _�K� Ker f / Gs���.
d�� (3.7)\� _�K� f��H ú� &ñ_��)a éß����<Êús���. ¢ N = Ker f�� ¿º���f(aNbN) = f(abN) = f(ab) = f(a)f(b) = aNbN
s�Ù¼�Ð f��H ï�r1lx+þAs���. f�� �����e���Ér ì�r"î �Ù¼�Ð f��H 1lx+þA�<Êús���.�� ���×e� 3.2.8 f : Z → Zn, a 7→ a��H Ker f = nZ��� �����ï�r1lx+þAs�Ù¼�Ð Z/nZ ' Zns���.
¢ det : GLn(R) → R∗, g 7→ det g��� �âĺ Ker det = SLn(R)s��¦ e��_�_� r ∈ R∗\� @/K� det(diag(r, 1, . . . , 1)) = rs�Ù¼�Ð GLn(R)/SLn(R) ' R∗s���.�� ���×e� 3.2.9 ï�r1lx+þA�<Êú ι : G→ AutG, g 7→ ιg_� �âĺ,
Ker ι = {a ∈ G | ιa(x) = x, ∀x ∈ G} = {a ∈ G | axa−1 = x, ∀x ∈ G} = Z(G)
s��¦ Im ι = InnGs�Ù¼�Ð G/Z(G) ' InnGs���.�� ��Ça�h� 3.2.3 (]j21lx+þA&ño�) H ≤ Gs��¦ N / Gs���� NH = HN ≤ Gs� 9H/H ∩N ' HN/Nh(H ∩N) 7→ hN
7£x"î. d�� (3.6)\� _�K� e��_�_� h ∈ H, n ∈ N\� @/K� hn = n′h ∈ NH, nh = hn′′ ∈HNs�Ù¼�Ð HN = NHs��¦ ����"f NH��H G_� ÂÒì�rç�Hs���. Óüt�:r N / Gs�Ù¼�Ð N /
NHs��¦ NH/N�Ér ç�Hs���. s�]jf : H → NH/N, f(h) = hN
s��� &ñ_� ����f(h1h2) = h1h2N = h1Nh2N
s�Ù¼�Ð f��H ï�r1lx+þAs���. Õª���X<NH/H = {hnN = hN | n ∈ N, h ∈ H}
s�Ù¼�Ð f��H �����s���. ¢Ker f = {h ∈ H | hN = N} = H ∩N
s�Ù¼�Ð &ño� 3.2.2(]j11lx+þA&ño�)\� _�K� H/H ∩N ' HN/Ns���.
40 ]j 3 �©� eç#�ÀÓü< eç#�ç�H�� ���×e� 3.2.10 G = Z, H = nZ, K = mZ�Ð ×þ� ���� H + K = nZ + mZ = (m,n)Z,
H ∩ K = [m,n]Zs�Ù¼�Ð nZ/[m,n]Z ' (m,n)Z/mZs���. #�l�"f (m,n) = gcd(m,n),
[m,n] = lcm(m,n)�¦ �����·p��. ½�&h���� \V�Ð n = 6, m = 8�Ð ú�ܼ��� 6Z/24Z '2Z/8Zs���.�� ���×e� 3.2.11
P = {(
a b0 d
)| a, d ∈ GL2(R), b ∈M2(R)} ⊂ GL4(R),
M = {(
a 00 d
)∈ P},
N = {(
1 b0 1
)∈ P}
���¦ ���. (a b0 d
) (a′ b′
0 d′
)=(
aa′′ ab′+bd′
0 dd′
)∈ P
s��¦ (a b0 d
)−1 =(
a −a−1bd1
0 d
)∈ P
s�Ù¼�Ð P��H GL4(R)_� ÂÒì�rç�Hs���.�� ��Ça�h� 3.2.4 (]j31lx+þA&ño�) N / G, K / G, N ≤ Ks���� K/N / G/Ns��¦
(G/N)/(K/N) ' G/K.
7£x"î. e��_�_� a ∈ G, k ∈ K\� @/K�
(aN)(kN)(aN)−1 = (aka−1)N ∈ K/N
s�ټ�РK/N / G/Ns���. s�]j
f : G/N → G/K, f(aN) = aK
���¦ &ñ_� ���. aN = bNs���� a−1b ∈ N ⊂ Ks�Ù¼�Ð d�� (3.3)\� _�K� aK = bKs���. ����"f f��H ú� &ñ_��)a �<Êús���. ì�r"îy� f��H �����ï�r1lx+þA�<Êús� 9
Ker f = {aN ∈ G/N | aK = K} = {aN | a ∈ K} = K/N
s�Ù¼�Ð &ño� 3.2.2(]j11lx+þA&ño�)\� _�K� (G/N)/(K/N) ' G/Ks���.
]j 3.2 ]X� &ñ½©ÂÒì�rç�Hõ� eç#�ç�H 41
�� ��Ça�h� 3.2.5 (7�£� Ça�h�) f : G→ G′�¦ �����ï�r1lx+þA�<Êú���¦ ����Φ : {H | Ker f ≤ H ≤ G} → {H ′ | H ′ ≤ G′}
H 7→ f(H)
�Ér {9�@/{9� @/6£xܼ�Ð"f G_� &ñ½©ÂÒì�rç�H�Ér G′_� &ñ½©ÂÒì�rç�Hõ� @/6£x�)a��.
7£x"î.
Ψ : {H ′ | H ′ ≤ G′} → {H | Ker f ≤ H ≤ G}
H ′ 7→ f−1(H ′)
s����¦ &ñ_� ��¦ Ψ ◦ Φ = I, Φ ◦Ψ = Ie���¦ 7£x"î ���� Φü< Ψ�� {9�@/{9� @/6£xe���¦ �Ðs���H �s� �)a��. z�]j�Ð f�� �����s�Ù¼�Ð e��_�_� H ′ ≤ G′\� @/K�
Φ ◦Ψ(H ′) = f(f−1(H ′)) = H ′
s��¦ d�� (3.8)\� _�K� Ker f ≤ H ≤ G��� e��_�_� H\� @/K�Ψ ◦ Φ(H) = f−1(f(H)) = ∪h∈Hf
−1(f(h)) = ∪h∈HhKer f = H ker f = H
s���. s�]j Ker f ≤ H / G�� ���� f�� �����s�Ù¼�Ð e��_�_� g′ ∈ G′\� @/K� f(g) =
g′��� g ∈ G�� �>rF� ��¦ g′f(H)(g′)−1 = f(g)f(H)f(g)−1 = f(gHg−1) = f(H)s�Ù¼�Ðf(H) / G′s���. %i�ܼ�Ð H ′ / G′s���� e��_�_� g ∈ G\� @/K�
f(gf−1(H ′)g−1) = f(g)f(f−1(H ′))f(g)−1 = f(g)H ′f(g)−1 = H ′
���X<gf−1(H ′)g−1 ⊃ Ker f
e��õ� Ψ,Φ�� {9�@/{9� @/6£xe���¦ s�6 x ����gf−1(H ′)g−1 = f−1(H ′)
s� ÷&Ù¼�Ð f−1(H ′) / Gs���.�� ���¿Ça�h� 3.2.12 N / Gs���� G/N_� ���H ÂÒì�rç�H�Ér H ≥ N��� G_� ÂÒì�rç�H H\� @/K�H/Ng1Js� 9 H/N / G/N{9� �9�¹Ø�æì�r�|��Ér H / Gs���.
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7£x"î. ³ðï�r�����ï�r1lx+þA pN : G → G/N�¦ Òqty�� ���� Ker pN = Ns��¦ H ⊃ Ker pN =
N��� G_� ÂÒì�rç�H H\� @/K� pN (H) = {hN | h ∈ H} = H/Ns�Ù¼�Ð 0A_� @/6£x&ño��ÐÂÒ'� s� &ño���H 7£x"î�)a��.�� ���×e� 3.2.13
V� 4 *�×
ËÂø5��»jÅ]���G�f!B�
43
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V� 4.1 â� ÒÏ��\��� ���¿Ça�h� 4.1.1 H,K�� ç�H G_� &ñ½©ÂÒì�rç�Hs��¦ H ∩ K = {e}s���� e��_�_� h ∈ H,
k ∈ K\� @/K� hk = khs���.
7£x"î.
hkh−1k−1 = (hkh−1)k−1 ∈ K
s��¦ ¢hkh−1k−1 = h(kh−1k−1) ∈ H
s���. Õª���X< H ∩K = {e}s�Ù¼�Ð hkh−1k−1 = es���. ����"f e��_�_� h ∈ H, k ∈ K\�@/K� hk = kh�� $íwn�ô�Ç��.
G = G1 ×G2�� ���.
G′1 = {(g1, e) | g1 ∈ G1} G′
2 = {(e, g2) | g2 ∈ G2}
�� ���� G′1 / G, G′
2 / Gs��¦ G′1 ∩G′
2 = {e}, G = G′1G
′2s� $íwn�ô�Ç��. %i�ܼ�Ð ç�H G�� ��
6£§�¦ ëß�7ᤠ���H ÂÒì�rç�H H,K\�¦ °ú���H���¦ ��&ñ ���:
(1) H,K / G,
(2) H ∩K = {e},(3) G = HK.
f : H ×K → G, f(h, k) = hk�� &ñ_� ���� &ño� 4.1.1\� _�K�f((h1, k1)(h2, k2)) = f(h1h2, k1k2) = h1h2k1k2 = h1k1h2k2 = f(h1, k1)f(h2, k2)
s�Ù¼�Ð f��H ï�r1lx+þAs���. Õª���X< f(h, k) = hk = es���� h = k−1 ∈ H ∩K = {e}s�Ù¼�Ðh = k = e�� ÷&#Q f��H éß���s���. ¢ G = HKs�Ù¼�Ð f��H �����s���. Õª�QÙ¼�Ð f��H 1lx+þA�<Êús��¦ G ' H ×Ks���. s� �âĺ, G\�¦ Hü< K_� 6�ÒÏ��\�(internal direct product)���¦ � 9 G = H ×K�Ð ��H��. s�\�¦ {9�ìøÍ�o �#��� ��K±ÓÞDº 4.1.2 ç�H G_� ÂÒì�rç�H H1,H2, . . . ,Hn\� @/K�
(1) ���H i = 1, 2, . . . , n\� @/K� Hi / G,
(2) G = H1H2 · · ·Hn,
(3) ���H i = 1, 2, . . . , n− 1\� @/K� (H1H2 · · ·Hi) ∩Hi+1 = {e}
]j 4.2 ]X� Ä»ô�ÇÒqt$í��6\�ç�H 45
�� $íwn� ���� G\�¦ H1,H2, . . . ,Hn_� ?/f��&h�s����¦ ô�Ç��.�� ���¿Ça�h� 4.1.3 ç�H G�� H1,H2, . . . ,Hn_� ?/f��&h�s���� G��H H1,H2, . . . ,Hn_� f��&h�õ� 1lx+þAs���.
7£x"î. ���$� &ño� 4.1.1\� _�K� i 6= js���� hi ∈ Hiü< hj ∈ Hj��H �§8�s� ��0px ���. ¢G = H1H2 · · ·Hns�Ù¼�Ð ���H g ∈ G��H
g = h1h2 · · ·hn (hi ∈ Hi)
+þAI��Ð ³ðr�÷&��HX< s��Qô�Ç ³ð�&³�Ér Ä»{9� ���. z�]j�Ð h1h2 · · ·hn = h′1h′2 · · ·h′n s����
h′nh−1n = (h′1)
−1h1(h′2)−1h2 · · · (h′n−1)
−1hn−1 ∈ Hn ∩H1H2 · · ·Hn−1 = {e}
s�Ù¼�Ð h′n = hns���. 0A d��\�"f hn = h′n�¦ ���ì�r ��¦ 0A �7HZO��¦ ìøÍ4�¤ ���� h′n−1 =
hn−1s� ÷&�¦, ) ±ú�&h�ܼ�Ð ���H i\� @/K� h′i = his���. s�]j
f : G→ H1 ×H2 × · · · ×Hn, f(h1h2 · · ·hn) = (h1, h2, . . . , hn)
s��� &ñ_� ���� f�� 1lx+þA�<Êúe���¦ ·ú� ú e����.
0A &ño��ÐÂÒ'� ?/f��&h�õ� f��&h��¦ ½ì�r �t� ·ú§�¦ G = H1 ×H2 × · · · ×Hns��� +��Áº~½Ó ���. f��&h�_� �|��¦ ¢-a�o �#��� ��K±ÓÞDº 4.1.4 ç�H G_� ¿º ÂÒì�rç�H N,H��
(1) N / G,
(2) H ∩N = {e},(3) G = HN
�¦ ëß�7ᤠ���� G\�¦ Nõ� H_� )K�ÒÏ��\�(semidirect product)���¦ � 9 G = N nH���¦ ��H��.
V� 4.2 â� ËÂø5��»jÅ]���G�f!B�G�� ËÂø5��»jÅ]���G�f!B�s�êøÍ Ä»ô�Ç>h_� G_� "é¶�è g1, g2, . . . , grs� �>rF�K�"f
G = 〈g1, g2, . . . , gr〉 = {n1g1 + n2g2 + · · ·+ nrgr}
46 ]j 4 �©� Ä»ô�ÇÒqt$í��6\�ç�H
s� ÷&��H ��¦ ú�ô�Ç��. s� �âĺ �<Êúf : Zr → G, f(n1, n2, . . . , nr) = n1g1 + n2g2 + · · ·+ nrgr
�Ér �����ï�r1lx+þAs���. f�� éß���{9� �9�¹Ø�æì�r�|��Érn1g1 + n2g2 + · · ·+ nrgr = 0 =⇒ n1 = n2 = · · · = nr = 0
���X<, s� �|�s� $íwn� ���� g1, g2, . . . , gr�¦ ����â���s��� ��¦ G = 〈g1, g2, . . . , gr〉_�e�$�(basis)\�¦ s�ê�r���¦ ô�Ç��. ¢ Ä»ô�Ç>h_� l�$�\�¦ °ú���H ��6\�ç�H�¦ ËÂø5��»jÅ]���ËÂ��G�f!B�(finitely generated free abelian group)s���ô�Ç��.Õª�QÙ¼�Ð G��Ä»ô�ÇÒqt$í��Ä»��6\�ç�H{9� �9�¹Ø�æì�r�|��Ér G ' Zr(1 ≥ r <∞)��� �s���. s� M:, r�¦ G_� N�ÊÁ(rank) ¢��H 'K��D���¦ ô�Ç��.�� ���¿Ça�h� 4.2.1 G��Ä»ô�ÇÒqt$í��Ä»��6\�ç�Hs���� ���Hl�$���H°ú �Ér>hú_�"é¶�è�Ðs�ÀÒ#Q�����.
7£x"î. G�� r>h_� "é¶�è�Ð s�ÀÒ#Q��� l�$�\�¦ °ú�ܼ��� G ' Zrs���. 2G = {2g | g ∈ G}�� ���� G/2G ' Zr/2Zr ' (Z/2Z)rs�Ù¼�Ð |G/2G| = 2rs��¦ G/2G��H l�$�ü< Áº�'a �Ù¼�ÐÄ»ô�Ç>h�Ð s�ÀÒ#Qt���H l�$�_� "é¶�è_� ú��H r�Ð Ä»{9� �>� ���&ñ�)a��.�� ���¿�Ça�h� 4.2.1 ëß���� {g1, · · · , gr}s� ��Ä»��6\�ç�H G_� l�$�s���� e��_�_� &ñú kü< i 6=j\� @/K� {g1, · · · , gj−1, gj + kgi, gj+1, · · · , gr}� G_� l�$�s���.�� ���¿Ça�h� 4.2.2 >�ú�� r��� Ä»ô�ÇÒqt$í��Ä»��6\�ç�H G_� e��_�_� ÂÒì�rç�H K��H >�ú�� r s� ���� Ä»ô�ÇÒqt$í��Ä»��6\�ç�Hs���. >����� {g1, · · · , gr}s� G_� l�$�s���� &h�{©�ô�Ç �ª�_� &ñúd1, · · · , ds�� �>rF�K�"f d1 | d2 | · · · | dss��¦ {d1g1, · · · , dsgs}�� K_� l�$��� �)a��.�� ��Ça�h� 4.2.1 ���H Ä»ô�ÇÒqt$í��6\�ç�H�Ér ��6£§_� +þAI�_� ç�Hõ� 1lx+þAs���:
Zm1 × · · · × Zmr × Z× · · · × Z
éß�, i = 1, . . . , r − 1\� @/K� mi | mi+1
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!B��+ ¼ÇУ� ø� Sylow Ça�h�
47
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V� 5.1 â� !B��+ ¼ÇУ� G\�¦ ç�Hs��� ��¦ X 6= {}\�¦ |9�½+Ës��� ���. ëß���� ��6£§_� �|��¦ ëß�7ᤠ���H �<Êú
ρ : G×X → X, (g, x) 7→ g · x
�� �>rF� ���� G�� (ρ\� ����) X0A\� ¼ÇУ� (act)ô�Ç��, ¢��H X\�¦ G-���TÒ¼s����¦ ��¦, ρ\�¦(X0A_�) G_� ���6 x(action)s����¦ ô�Ç��:
(1) e��_�_� x ∈ X\� @/K� ρ(e, x) = x,
(2) e��_�_� a, b ∈ G, x ∈ X\� @/K� ρ(a, ρ(b, x)) = ρ(ab, x). ë�HÐ�o�©� ρ�� Áº%Ás�t� SX�z�ô�Ç �âĺ ρ(a, x)\�¦ éß�í�Hy� a · x ¢��H ax���¦ ��H��.
y�� g ∈ G\� @/K� ρg : X → X, x 7→ g · x���¦ &ñ_� ���� 0A_� &ñ_���H(1) ρe = IXs� 9(2) e��_�_� a, b ∈ G\� @/K� ρaρb = ρab�� $íwn�ô�Ç����H �s���.
:£¤y� e��_�_� a ∈ G\� @/K� ρaρa−1 = ρa−1ρa = IXs�Ù¼�Ðρa ∈ SX
s��¦θρ : G→ SX , g 7→ ρg
���¦ &ñ_� ���� θρ�Ér ï�r1lx+þA�<Êú�� �)a��. %i�ܼ�Ð e��_�_� ï�r1lx+þA�<Êú θ : G → SX�� ÅÒ#Qt����
ρ = ρθ : G×X → X, (g, x) 7→ θ(g)(x)
��H G_� ���6 xs� �)a��. z�]j�Ðρ(e, x) = θ(e)(x) = IX(x) = x
s��¦ρ(a, ρ(b, x)) = θ(a)(θ(b)(x)) = θ(ab)(x) = ρ(ab, x)
s���. ¢, ÅÒ#Q��� G_� ���6 x ρ\� @/K�ρθρ(g, x) = θρ(g)(x) = ρg(x) = ρ(g, x)
s��¦ ÅÒ#Q��� ï�r1lx+þA�<Êú θ : G→ SX\� @/K�θρθ
(g)(x) = (ρθ)g(x) = ρθ(g, x) = θ(g)(x)
s���. Õª�QÙ¼�Ð ��6£§_� &ño�\�¦ %3���H��.
]j 5.1 ]X� ç�H_� ���6 x 49
�� ���¿Ça�h� 5.1.1 X0A_� G_� ���6 x[þt_� |9�½+Ë�¦ A, G\�"f SX�Ð ����H ï�r1lx+þA�<Êú[þt_� |9�½+Ë�¦ B�� ���. @/6£x ρ 7→ θρ��H Aü< B��s�_� ú� &ñ_��)a ���éß��� �<Êú�Ð"f Õª %i��<Êú��Hθ 7→ ρθ�Ð ÅÒ#Q�����.
7£x"î.
s� &ño�\� _�K� X0A\� G_� ���6 xs�êøÍ G�ÐÂÒ'� SX�Ð_� ï�r1lx+þA�<Êú���¦ Òqty��K���)a��.
X0A\� ��6£§õ� °ú s� ô�Ç �'a>�\�¦ &ñ_� ���:
x ∼ x′ ⇐⇒ ∃g ∈ G(x′ = gx).
Õª�Q��� x = exs��¦, x′ = gxs���� x = g−1x′s� 9, x′ = gx, x′′ = hx′s���� x′′ = hgxs�Ù¼�Ð ∼�Ér 1lxu��'a>�s���. s� 1lxu��'a>�\� _�ô�Ç x_� 1lxu�ÀÓ [x]\�¦ x_� F��¿���¦ � 9[x] = Gx���¦ ³ðr�ô�Ç��. Õª�QÙ¼�Ð X��H "f�Ð ���Ér C��[þt_� "f�Ð �è��� ½+Ë|9�½+Ës���. 7£¤,
X0 = {x ∈ X | ∀g ∈ G(gx = x)}
�� ���� x ∈ X0{9� �9�¹Ø�æì�r�|��Ér Gx = {x}��� �s�Ù¼�Ð X\�¦X = X0 ∪Gx1 ∪ · · · ∪Gxn, |Gxi| > 1 (5.1)
%�!3� "f�Ð �è��� |9�½+Ë_� ½+Ë|9�½+Ëܼ�Ð jþt ú e����.
s�]j y�� x ∈ X\� @/K�Gx = {g ∈ G | gx = x}
�� ���. ex = xs�Ù¼�Ð e ∈ Gxs��¦, a, b ∈ Gxs���� bx = xs�Ù¼�Ð x = b−1xs��¦ ����"fab−1x = ax = xs���.Õª�QÙ¼�Ð Gx��H G_�ÂÒì�rç�Hs���. Gx\�¦ x_��§Ça�ÉÙ&P�!B�(subgroup
fixing x), ò5ÑÇa�ÉÙ&P�!B�(stabilizer) ¢��H �� ��ÉÙ&P�!B�(isotropy subgroup)s����¦ ô�Ç��.�� ���×e� 5.1.2 ��6£§�Ér �¿º ���6 x_� \Vs���.
• Sn×{1, 2, . . . , n} → {1, 2, . . . , n}, (σ, i) 7→ σ(i). s��âĺ i_��¦&ñÂÒì�rç�H�Ér (Sn)i =
{σ ∈ Sn | σ(i) = i} ' Sn−1s� 9 i_� C����H (Sn)(i) = {1, 2, . . . , n}s���.
• H ≤ G{9� M:, H ×G→ G, (h, g) 7→ hg. s� ���6 x�¦��¥�l��â (left translation)s����¦ ô�Ç��. s� �âĺ �¦&ñÂÒì�rç�H�Ér Hg = {e}s��¦ g_� C����H Hg, 7£¤, g_� eç#�ÀÓs���. K�� ¢ ���Ér G_� ÂÒì�r|9�½+Ës���� H × G/K → G/K, h(xK) 7→ hxK� ���6 xs���.
50 ]j 5 �©� ç�H_� ���6 xõ� SYLOW &ño�
• H ≤ G{9�M:, H×G→ G, (h, g) 7→ hgh−1. s����6 x�¦ �»bËc¼ÇУ� s����¦ô�Ç��.s��âĺ �¦&ñÂÒì�rç�H�Ér Hg = {h ∈ H | gh = hg}s� 9 C����H Hg = {hgh−1 | h ∈ H}s���. :£¤y� H = G{9� M:, Hg\�¦ g_� ä»ÐÏ��ªÉÙ&P�!B�(centralizer)s��� ��¦
CG(g) = {a ∈ G | ag = ga}
���¦ ����?/ 9, g_� C��gG = {aga−1 | a ∈ G}
\�¦ g_� �»bËcÇÚ(conjugate class)���¦ ô�Ç��. ¢ X\�¦ G_� ���H ÂÒì�rç�H[þt_� |9�½+Ës����¦ ���� H ×X → X, (h,K) 7→ hKh−1� ���6 xs���. hKh−1�¦ K_� /BNÓ�os���ô�Ç��.s�M:, K_��¦&ñ�oÂÒì�rç�H�¦ H\�"f K_�Ça�Ī�ªÉÙ&P�!B�(mormalizer)���¦ � 9 NH(K)�Ð ����?/ 9 H = G��� �âĺ_�
NG(K) = {g ∈ G | gKg−1 = K}
\�¦ éß�í�Hy� K_� &ñ½©�oÂÒì�rç�Hs��� ô�Ç��. Õª�Q��� {©����y� K / NG(K)s� 9, ¢ K /
G{9� �9�¹Ø�æì�r�|��Ér NG(K) = G��� �s���.�� ���¿Ça�h� 5.1.3 (��È*�×ÛÖS Cayley Ça�h�) [G : H] = ms���� ker θ ≤ H��� ï�r1lx+þA �<Êúθ : G→ Sms� �>rF�ô�Ç��.
7£x"î. X = G/H0A\�"f G_� ýa8£¤s�1lx ���6 x�¦ Òqty�� ��¦ &ño� 5.1.1�¦ &h�6 x ���� θ1 :
G→ SX , θ1(g)(xH) = gxH\�¦ %3���H��. s� M:,
ker θ1 = {g ∈ G | gxH = xH, ∀x ∈ G}
= {g ∈ G | x−1gx ∈ H ∀x ∈ G}
= {g ∈ G | g ∈ xHx−1, ∀x ∈ G}
= ∩x∈GxHx−1 ≤ H
s���. s�]j θ1õ� 1lx+þA�<Êú SX ' Sm�¦ ½+Ë$í ���� &ño�_� θ\�¦ %3���H��.�� ���¿Ça�h� 5.1.4 y�� x\� @/K� �<Êú Gx → G/Gx, gx 7→ gGx��H x_� C��\�"f Gx_� ýa8£¤eç#�ÀÓ_� |9�½+Ëܼ�Ð ����H ���éß����<Êús���. 7£¤, |Gx| = [G : Gx]s���.
7£x"î. e��_�_� a, b ∈ G\� @/K�ax = bx ⇐⇒ a−1bx = x ⇐⇒ a−1b ∈ Gx ⇐⇒ aGx = bGx
s�Ù¼�Ð 0A �<Êú��H ú� &ñ_��)a ���éß����<Êús���.
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�� ������Ça�h� 5.1.5 K\�¦ Ä»ô�Çç�H G_� ÂÒì�rç�Hs��� ����(1) |gG| = [G : CG(g)]s� 9 ����"f s���Ér |G|_� ���ús���.
(2) gG1 , g
G2 , . . . , g
Gn\�¦ G_� "f�Ð ���Ér /BNÓ�oÀÓ �������¦ ����
|G| =n∑
i=1
[G : CG(gi)] (5.2)
s� $íwn�ô�Ç��. s� d���¦ G_� ÇÚ'�×Ça�ÐÏ�(class equation)s��� ô�Ç��.
(3) K_� /BNÓ�oÂÒì�rç�H_� >hú��H [G : NG(K)]s� 9 |G|_� ���ús���.
ô�Ǽ#� a ∈ Z(G){9��9�¹Ø�æì�r�|��ÉrG = CG(a), ¢��H aG = {a}s�Ù¼�Ð gG1 , g
G2 , . . . , g
Gm\�¦
"é¶�è_� ú�� 2 s��©���� "f�Ð ���Ér /BNÓ�oÀÓ ������ ���� 0A_� ÀÓ~½Ó&ñd���Ér
|G| = |Z(G)|+m∑
j=1
[G : CG(gj)] (5.3)
���¦ jþt ú e����.�� ���×e� 5.1.6 Sn_� ÀÓ~½Ó&ñd��.
{9�ìøÍ&h�ܼ�Ð ç�H G�� |9�½+Ë X\� ���6 x ���H �âĺ, d�� (5.1)ܼ�ÐÂÒ'�|X| = |X0| ∪ |Gx1| ∪ · · · ∪ |Gxn|, |Gxi| > 1 (5.4)
s� $íwn�ô�Ç��. s� d���¦ ��ð5��ªÛÖS ÇÚ'�×Ça�ÐÏ�s��� ô�Ç��.�� ���¿Ça�h� 5.1.7 p�� ç�H G_� 0Aú\�¦ ��¾º��H ���©� ����Ér �èú�� ���� [G : H] = p��� ���HÂÒì�rç�H H��H &ñ½©s���.
7£x"î. X = G/H�� Z�~ܼ��� SX ' Sps���. ç�H G�Ð X\� ýa8£¤s�1lx ���6 x�¦ �#� s�\� @/6£x ���H ï�r1lx+þA τ : G→ SX\�¦ Òqty�� ��¦ K = Ker τ�� ���. 7£¤, e��_�_� g, x ∈ G\� @/K�τ(g)(xH) = gxHs���. g ∈ Ks���� ���H x\� @/K� gxH = xHs��¦, :£¤y� gH = Hs�Ù¼�Ð K ⊂ Hs���. ]j11lx+þA&ño�\� _�K� G/K ' Im τ ≤ SX ' Sps�Ù¼�Ð Lagrange&ño�\�_�K� |G/K| | p!s���. Õª���X< |G/K|��H |G|_� ���ús��¦ p�� |G|_� ���©� ����Ér �èú��� ���ús�Ù¼�Ð |G/K|��H 1 ¢��H ps�#Q�� ���
|G/K| = [G : K] = [G : H][H : K] = p[H : K] ≥ p
s�Ù¼�Ð |G/K| = ps���. ����"f [H : K] = 1s�Ù¼�Ð H = K��H &ñ½©s���.
52 ]j 5 �©� ç�H_� ���6 xõ� SYLOW &ño�
V� 5.2 â� Sylow Ça�h��� ���¿�Ça�h� 5.2.1 0Aú�� pn(p��H �èús��¦, n ≥ 1)��� ç�H G�� Ä»ô�Ç|9�½+Ë X\� ���6 xô�Ç�� ��¦ X0 = {x ∈ X | ∀g ∈ G(gx = x)}�� ���� |X0| ≡ |X| (mod p)s���.
7£x"î. x ∈ X0{9� �9�¹Ø�æì�r�|��Ér Gx = {x}��� �s�Ù¼�Ð X\�¦X = X0 ∪Gx1 ∪ · · · ∪Gxn, (|Gxi| > 1)
%�!3� "f�Ð �è��� |9�½+Ë_� ½+Ë|9�½+Ëܼ�Ð jþt ú e����. Õª�QÙ¼�Ð|X| = |X0|+ |Gx1|+ · · ·+ |Gxn|
���X< ���H i\� @/K�1 < |Gxi| = [G : Gxi ] | |G| = pn
s�Ù¼�Ð p | |Gxi|s���. ����"f |X| ≡ |X0| (mod p)s���.�� ��Ça�h� 5.2.1 (Cauchy�+ Ça�h�) ëß���� �èú p�� ç�H G_� 0Aú\�¦ ��¾º��� G��H 0Aú�� p���ÂÒì�rç�H�¦ °ú���H��.
7£x"î.
X = {(a1, a2, . . . , ap) | ai ∈ G, a1a2 · · · ap = e}
�� ����ap = (a1a2 · · · ap−1)−1
�Ð ���&ñ÷&Ù¼�Ð|X| = |G|p−1 ≡ 0 (mod p)
s���. ô�Ǽ#� a1a2 · · · ap = es���� e��_�_� k = 1, 2, . . . , n− 1\� @/K�ak+1ak+2 · · · apa1 · · · ak = ak+1ak+2 · · · ap(ak+1ak+2 · · · ap)−1 = e
s�ټ�РZp�РX\�k(a1, a2, . . . , an) = (ak+1, ak+2, . . . , ak)
õ� °ú s� í�H8����6 x�¦ ½+É Ãº e����. Õª���X< (e, e, . . . , e) ∈ X0s�Ù¼�Ð |X0| > 0s��¦ 0A �Ð�&ño�\� _�K� |X0| ≡ |X| ≡ 0 (mod p)s�Ù¼�Ð |X0| ≥ ps���. ����"f a 6= e��� "é¶�è a���>rF�K�"f (a, a, . . . , a) ∈ X0s��¦ s���H |a| = pe���¦ _�p� �Ù¼�Ð G��H 0Aú�� p��� ÂÒì�rç�H〈a〉\�¦ �í�<Êô�Ç��.
]j 5.2 ]X� SYLOW &ño� 53
�� ��K±ÓÞDº 5.2.1 p\�¦ �èú�� ���. ëß���� ç�H G_� ���H "é¶�è_� 0Aú�� p_� ��[þv]jY�L g1Js����G\�¦ p-!B�s��� ô�Ç��. G�� e��_�_� ç�Hs��¦ H ≤ G�� p-ç�Hs���� H\�¦ G_� p-ÉÙ&P�!B�s��� ô�Ç��. G��Ä»ô�Çç�Hs��¦ pn | |G|, pn+1 - |G|, n > 0{9�M:,0Aú�� pm��� G_�ÂÒì�rç�H�¦ Sylow
p-ÉÙ&P�!B�s��� ô�Ç��.
G�� Ä»ô�Çç�H��� �âĺ��H Lagrange&ño�ü< Cauchy&ño�\� _�K� G�� p-ç�H{9� �9�¹Ø�æì�r�|��Ér |G|�� p_� ��[þv]jY�L g1J��� �s���. ¢ H�� G_� Sylow p-ÂÒì�rç�Hs���� H_� ���H /BNÓ�o� Sylow p-ÂÒì�rç�Hs� 9, ëß���� G�� Ä»{9�ô�Ç Sylow p-ÂÒì�rç�H P\�¦ ��t���� P / Gs���.�� ���¿Ça�h� 5.2.2 G 6= {e}�� p-ç�Hs���� Z(G) 6= {e}s���.
7£x"î. ÀÓ~½Ó&ñd�� (5.3)\� _� ����
|G| = |Z(G)|+m∑
j=1
[G : CG(gj)], ([G : CG(gj)] > 1)
���X< [G : CG(gj)] | |G| = pks�Ù¼�Ð ���H j\� @/K� p | [G : CG(gj)]s���. ����"fp | |Z(G)|s�Ù¼�Ð |Z(G)| ≥ ps���.�� ���¿Ça�h� 5.2.3 |G| = p2(p��H �èú)s���� G��H ��6\�ç�Hs���.
7£x"î. Z(G) 6= {e}s�Ù¼�Ð |Z(G)| = p ¢��H p2s���. |Z(G)| = p�� ���� |G/Z(G)| = ps�Ù¼�Ð G/Z(G)��H í�H8�ç�Hs� ÷&#Q G/Z(G) = 〈gZ(G)〉��� g�� �>rF�ô�Ç��. Õª�Q��� a, b ∈ G\�@/K� x, y ∈ Z(G)�� �>rF�K�"f a = gix, b = gjy�� ÷&Ù¼�Ð ab = gixgjy = gjygix = ba��$íwn� ��¦ G��H ��6\�ç�Hs� ÷&#Q Z(G) = G�� ÷&Ù¼�Ð |Z(G)| = p��� �âĺ��H \O���. ����"f|Z(G)| = p2, 7£¤ G = Z(G)s�Ù¼�Ð G��H ��6\�ç�Hs���.
Ä»ô�Çç�H G_� ÂÒì�rç�H H��H ýa8£¤eç#�ÀÓ_� |9�½+Ë X = G/H\� ýa8£¤s�1lxܼ�Ð ���6 xô�Ç��.
s� �âĺ
xH ∈ X0 ⇐⇒ ∀h(hxH = xH) ⇐⇒ ∀h(x−1hx ∈ H) ⇐⇒ x ∈ NG(H)
s�Ù¼�Ð |X0| = [NG(H) : H]s���. ����"f ëß���� H�� p-ÂÒì�rç�Hs���� �Ð�&ño� 5.2.1\� _�K��� ���¿�Ça�h� 5.2.2 Ä»ô�Çç�H G_� ÂÒì�rç�H H\� @/K� [NG(H) : H] ≡ [G : H] (mod p)�� $íwn� � 9 �8ç�H���� p | [G : H]s���� NG(H) > Hs���.
54 ]j 5 �©� ç�H_� ���6 xõ� SYLOW &ño��� ��Ça�h� 5.2.2 (V�1 Sylow Ça�h�) p�� �èús��¦, |G| = pnm (n ≥ 1, (p,m) = 1)s�������H 1 ≤ i ≤ n\� @/K� 0Aú�� pi��� G_� ÂÒì�rç�Hs� &h�#Q� ��� �>rF� ��¦, 0Aú��pi(1 ≤ i < n)��� y�� ÂÒì�rç�H�Ér 0Aú�� pi+1��� #Q�"� ÂÒì�rç�H_� &ñ½©ÂÒì�rç�Hs� �)a��.
7£x"î. p | |G|s�Ù¼�Ð Cauchy_� &ño�\� _�K� G��H 0Aú�� p��� ÂÒì�rç�H�¦ °ú���H��. s�]j ) ±ú�&h�ܼ�Ð G�� 0Aú�� pi(1 ≤ i < n)��� ÂÒì�rç�H H\�¦ °ú���H���¦ ���. Õª�Q���
1 < |NG(H)/H| ≡ [G : H] (mod p)
s�Ù¼�Ð p | |NG(H)/H|s���. ����"f NG(H)/H��H 0Aú�� p��� ÂÒì�rç�H H1/H\�¦ °ú���HX<,
Õª�Q��� |H1| = p|H| = pi+1s��¦ H /H1s� �)a��.�� ��Ça�h� 5.2.3 (V�2 Sylow Ça�h�) H�� Ä»ô�Çç�H G_� p-ÂÒì�rç�Hs��¦ P�� G_� Sylow p-ÂÒì�rç�Hs���� &h�{©�ô�Ç g ∈ G�� �>rF�K�"f gHg−1 ≤ Ps���. :£¤y�, e��_�_� ¿º Sylow p-ÂÒì�rç�H�Ér /BNÓ�os� 9 ����"f P / G{9� �9�¹Ø�æì�r�|��Ér P�� G_� Ä»{9�ô�Ç Sylow p-ÂÒì�rç�H��� �s���.
7£x"î. H��H |9�½+Ë X = G/P\� ýa8£¤s�1lxܼ�Ð ���6 xô�Ç��. |G| = pnm((p,m) = 1)s��� ���. Õª�Q��� |P | = pn, |X| = [G : P ] = ms�Ù¼�Ð p - |X|s���. Õª���X< �Ð�&ño� 5.2.1\�_�K�
|X| ≡ |X0| (mod p)
s�Ù¼�Ð p - |X0|s��¦ :£¤y� X0 6= {}s���. Õª�QÙ¼�Ð g ∈ X0�� �>rF�K�"f ���H h ∈ H\� @/K� hgP = gPs���. ����"f g−1Hg ⊂ P , ¢��H H ≤ gPg−1s���.�� ��Ça�h� 5.2.4 (V�3 Sylow Ça�h�) |G| = pnm((p,m) = 1)s��¦ G_� Sylow p-ÂÒì�rç�H_� >hú\�¦ np�� ����
(1) e��_�_� Sylow p-ÂÒì�rç�H P\� @/K� np = [G : N(P )],
(2) np ≡ 1 (mod p),
(3) np | m.
7£x"î. X\�¦ G_� ���H Sylow p-ÂÒì�rç�H_� |9�½+Ës��� ��¦ P ∈ X\�¦ �¦&ñ ���. ]j2 Sylow
&ño�\� _�K� ���H Sylow p-ÂÒì�rç�H�Ér "f�Ð /BNÓ�os�Ù¼�Ð &ño� 5.1.5(3)\� _�K� np = [G :
N(P )]s���. s�]j P�Ð X0A\� /BNÓ�o���6 x�¦ ���� �Ð�&ño� 5.2.1\� _�K�np = |X| ≡ |X0| (mod p)
]j 5.2 ]X� SYLOW &ño� 55
s���. Õª���X<X0 = {Q ∈ X | pQp−1 = Q, ∀p ∈ P}
s�Ù¼�Ð P ∈ X0e���Érì�r"î ��¦,ëß���� Q ∈ X0s���� P ≤ N(Q)s�Ù¼�Ð Pü< Q �¿º N(Q)_�Sylow p-ç�Hs� ÷&��HX<, Q / N(Q)s�Ù¼�Ð ]j2 Sylow &ño�\� _�K� P = Qs���. Õª�QÙ¼�Ð X0 = {P}s��¦ ����"f np ≡ 1 (mod p)s���. ��t�}��ܼ�Ð np | pnm���X< np ≡ 1
(mod p)s�Ù¼�Ð (np, p) = 1s� ÷&#Q np | ms� $íwn�ô�Ç��.�� ���×e� 5.2.4 |G| = pq(p, q �èú)s���� G��H éß�í�Hç�Hs� ��m���.
7£x"î. p = qs���� G��H0Aú�� p2�����6\�ç�Hs�Ù¼�Ðéß�í�Hç�Hs���m���. p > qs����]j3 Sylow
&ño�\� _�K� np ≡ 1 (mod p)s��¦ np | q���X< p > qs�Ù¼�Ð np = 1s���. ]j2 Sylow &ño�\� Sylow p-ÂÒì�rç�H�Ér &ñ½©s���.�� ���×e� 5.2.5 |G| = 175 = 52 · 7��� ç�H G��H ��6\�ç�Hs���.
7£x"î. n5 ≡ 1 (mod 5)s��¦ n5 | 7s�Ù¼�Ð n5 = 1s���. ����"f G��H Ä»{9�ô�Ç &ñ½©��� Sylow
5-ÂÒì�rç�H P\�¦ °ú��¦ P / Gs���. ��ðøÍ��t��Ð n7 ≡ 1 (mod 7), n7 | 25s�Ù¼�Ð n7 = 1s�#Q"f G��H Ä»{9�ô�Ç ����"f &ñ½©��� Sylow 7-ÂÒì�rç�H Q\�¦ °ú���H��. |P | = 25, |Q| = 7s�Ù¼�ÐP ∩Q = {e}s��¦, ����"f G = PQ =' P ×Q���X< P,Q�� ��6\�ç�Hs�Ù¼�Ð G� ��6\�ç�Hs���.�� ���×e� 5.2.6 |G| = 56 = 23 · 7s���� G��H éß�í�Hç�Hs� ��m���.
7£x"î.�� ���×e� 5.2.7 |G| = 72 = 23 · 32s���� G��H éß�í�Hç�Hs� ��m���.
7£x"î. ]j3 Sylow&ño�\�_�K� n3 = 1, 4s���. P\�¦ô�Ç Sylow 3-ÂÒì�rç�Hs��� ���. n3 = 1s���� P / Gs���. n3 = 4s���� ]j3 Sylow &ño�\� _�K� [G : N(P )] = n3 = 4s�Ù¼�Ð SX��©��)a Cayley &ño�\� _�K� ï�r1lx+þA�<Êú θ : G → S4, ker θ ≤ N(P )�� �>rF�ô�Ç��. Õª���X<|G| = 72 > 24 = |S4s�Ù¼�Ð | ker θ| 6= 1s�#Q"f ker θ��H G_� ��"î �t� ·ú§�Ér &ñ½©ÂÒì�rç�Hs���.
s�]j ç�H_� ���6 xs� !lr\� #Qb�G>� 6£x6 x÷&��Ht� ·ú����Ðl� 0AK� Ä»ô�Çç�H G�� Ä»ô�Ç|9�½+ËX0A\� ���6 xô�Ç���¦ ���. e��_�_� g ∈ G\� @/K�
Xg = {x ∈ X | gx = x}
���¦ Z�~��.
56 ]j 5 �©� ç�H_� ���6 xõ� SYLOW &ño��� ��Ça�h� 5.2.5 (Burnside �»ÐÏ�) r�¦ G_� ���6 x\� _�ô�Ç X_� C��_� >hú�� ����
r =1|G|
∑g∈G
|Xg|.
7£x"î. Y = {(g, x) ∈ G×X | gx = x}, N = |Y |�� ���. �¦&ñ�)a g ∈ G\� @/K�|{(g, x) ∈ Y | x ∈ X}| = |Xg|
s�Ù¼�Ð N =∑
g∈G |Xg|s���. ô�Ǽ#� �¦&ñ�)a x ∈ X\� @/K�|{(g, x) ∈ Y | g ∈ G}| = |Gx|
s�Ù¼�ÐN =
∑x∈X
|Gx| =∑x∈X
|G||Gx|
= |G|∑x∈X
1|Gx|
s�l�� ���. Õª���X< X_� e��_�_� C�� O\� @/K� ∑x∈O
1|Gx| = 1s�Ù¼�Ð N = |G|rs� ÷&
#Q &ño���H 7£x"î÷&%3���.