Math. Ann. 315, 503–510 (1999) MathematischeAnnalenc© Springer-Verlag 1999

Putting the squeeze on the Noether gap– The case of the alternating groupsAn

Larry Smith 1,2

1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA2 Mathematisches Institut der Universitat, D-37073 Gottingen, Germany

Received: 15 June 1998 / in final form: 3 January 1999

Abstract. Let ρ : G ↪→ GL(n, F) be a faithful representation of thefinite groupG over the fieldF. In 1916 E. Noether proved that forF ofcharacteristic zero the ring of invariantsF[V ]G is generated as an algebraby the invariant polynomials of degree at most|G|. This upper bound forthe degrees of the generators in a minimal generating set is referred to asNoether’s bound. The result has been generalized to the case where thecharacteristic ofF is greater than the order ofG, or when the characteristicof F is prime to the order ofG, and the groupG is solvable. In this note weprove that if Noether’s bound fails when the order of the group is prime tothe characteristic ofF, referred to as the nonmodular case, then it fails fora finite nonabelian simple group. We then show how yet another reworkingof Noether’s argument leads to a proof that Noether’s bound holds in thenonmodular case for the alternating groups.

Mathematics Subject Classification (1991):13A50

Let G be a finite group andρ : G ↪→ GL(n, F) a faithful representationof G over the fieldF. Via ρ the groupG acts on the vector spaceV = F

n,and hence also on the graded algebraF[V ] of homogeneous polynomialfunctions onV . The subalgebra of functionsF[V ]G fixed by the action ofG is called thering of invariants of G. As a general reference on invarianttheory see [3].

In [2] E. Noether proved that whenever the characteristic ofF is zerothenF[V ]G is a finitely generated algebra overF. In addition she gave analgorithm to construct a system of generators using polarizations of ele-mentary symmetric polynomials (see also [6] VIII.B.15 or [3] Sect. 3.3).From this she deduced thatF[V ]G is generated by invariant polynomials of

504 L. Smith

degree at most the order ofG, which is denoted by|G|. This upper boundfor the degrees of the generators in a minimal generating set is referred toasNoether’s bound.

If the ground fieldF has characteristicp, then following standard prac-tice we call the representationρ : G ↪→ GL(n, F) nonmodular if p doesnot divide|G| andmodular otherwise. Noether’s bound is known to holdif p > |G| (see [5] or [3] Theorem 3.1.10), for solvable groups in the non-modular case [4], and it has been conjectured to hold for all nonmodularrepresentations. It may fail (see for example [3] Sect. 4.2 Example 2) ifpdivides |G|. The region,p < |G|, but p 6 | |G|, where Noether’s bound isnot known either to hold or to fail, is theNoether gapof the title. The firstgoal of this note is to show that if Noether’s bound fails in the nonmodularcase, then it fails for a finite nonabelian simple group. We then show howyet another reworking of one of Noether’s argument leads to a proof for thealternating groups that Noether’s bound holds in the nonmodular case.

This research was done while preparing a lecture on Noether’s boundfor a Crash Course on Invariant Theory given by the author as OrdwayVisitor at the School of Mathematics of the University of Minnesota. I wouldlike to thank the more than twenty participants in this lecture series for theirattentive and critical attitude, and probing questions.

1. Reducing the Noether gap to simple groups

Let us begin by introducing some notation. IfA is a graded connected com-mutative Noetherian algebra over a fieldF, denote byβ(A) the maximumdegree of a (homogeneous) generator ofA in any minimal generating set forA as anF algebra. This is nothing but the degree of the Poincare polynomialP (QA, t) of the module of indecomposablesQA := F ⊗A A = A/(A)2,whereA ⊂ A denotes the augmentation ideal. (See [3] Chapter 4 and Theo-rem 5.2.5.) Henceβ(A) does not depend on the choice of minimal generatingset. IfG is a finite group andρ : G ↪→ GL(n, F) a faithful representationof G over the fieldF, then we denoteβ(F[V ]G) by β(ρ). We set

βF(G) = max {β(ρ) | ρ : G ↪→ GL(n, F)} .

The following proposition is proved by a simple linearization process as in[4].

Proposition 1.1. Let A be a graded connected commutative algebra overa fieldF andG ≤ Aut∗(A) a finite group of grading preserving automor-phisms ofA. If |G| ∈ F

× then

β(AG) ≤ β(A)βF(G) .

Putting the squeeze on the Noether gap 505

Proof. Let V =β(A)⊕i=1

Ai, whereAi denotes the homogeneous component

of A of degreei. The groupG acts onV by linear automorphisms, and thenatural map

S(V ) −→ A

is aG-equivariant epimorphism, whereS(V ) denotes the symmetric algebraon the graded vector spaceV . (Apart from the grading onV this is justF[V ∗],whereV ∗ is the dual ofV .) Since the characteristic ofF does not divide theorder ofG the induced map

S(V )G −→ AG

is also onto, and, taking account of the maximum degree of an element ofV , yields the result. utCorollary 1.2. LetG be a finite group,N / G a normal subgroup, andF afield of characteristicp. Supposeρ : G ↪→ GL(n, F) is a representation ofG overF andρ |N the restriction ofρ to N . ThenG/N acts onF[V ]N and,if p 6 | |G|, then

β(ρ) ≤ β(ρ |N )βF(G/N) ,

and henceβF(G) ≤ βF(N)βF(G/N) .

Proof. This follows from the isomorphism (see [3] Proposition 1.5.1)

F[V ]G ∼= (F[V ]N)G/N

and Proposition 1.1. utCorollary 1.3. LetG be a finite group andF a field. Suppose the character-istic ofF does not divide the order ofG. If G has a composition series whosecomposition factorsK1, . . . , Km satisfyβF(Ki) ≤ |Ki|, for i = 1, . . . , m,thenβF(G) ≤ |G|.Proof. This follows from Corollary 1.2 by induction onm. utCorollary 1.4. LetF be a field of characteristicp. If there is a finite groupG withp 6 | |G| for whichβF(G) > |G|, then there is also a finite nonabeliansimple groupS with βF(S) > |S|.Proof. ChooseG of minimal order such thatp 6 | |G| andβF(G) > |G|. ThenG is simple. For, if not, we may choose a nontrivial normal subgroupN /G.Since|N |, |G/N | < |G| it follows that βF(N) < |N | andβF(G/N) <|G/N |, whence from Corollary 1.2 we obtain

βF(G) ≤ βF(N)βF(G/N) ≤ |N | · |G/N | = |G| ,

which would be a contradiction. HenceG is simple, and by [4] Lemma 1 itis not cyclic of prime order, so must be nonabelian.ut

506 L. Smith

2. Closing the Noether gap for the alternating groups

We begin with a generalization of [3] Theorem 2.4.2. For this we requiresome terminology. Suppose thatC ⊆ A is a finite integral extension ofgraded connected commutative Noetherian algebras over a field. Ifa ∈ Athena is a root of a monic polynomial of minimal degreema(X) ∈ C[X],called theminimal polynomial of a over C. We set

deg(A|C) = max {deg(ma(X)) | a ∈ A}and call it thedegree ofA over C. As usual, if there is no maximum, wesetdeg(A|C) = ∞.

Recall the following combinatorial lemma from [3] Lemma 2.4.1.

Lemma 2.1. LetV be a vector space over a fieldF andu1, . . . , uj ∈ F[V ].If j! 6= 0 ∈ F then the monomialu1 · · ·uj is a linear combination ofj-thpowers of sums of elements of{u1, . . . , uj}.

Proof. This follows from the formula

(−1)jj !u1 · · ·uj =∑

I⊆{1,...,j}(−1)|I|

(∑i∈I

ui

)j

.

In this formula,I runs over all subsets of{1, . . . , j} and|I| is the cardinalityof I. utTheorem 2.2. Let C ⊆ A be a finite integral extension of graded con-nected commutative Noetherian algebras over the fieldF of characteristicp. Suppose

(i) deg(A|C) is finite,(ii) p > deg(A|C), and(iii) there exists aC-module splitting

π : A −→ C

to the inclusionC ↪→ A.

Thenβ(C) ≤ β(A) · deg(A|C).

Proof. The proof is a minor modification of the proof of [3] Theorem 2.4.2,and as much as possible we employ the same notations.

Set β(A) = m and deg(A|C) = d. Let B be the subalgebra ofCgenerated by elements of degree at mostmd. Our goal is to show thatB = C. To this end introduce

N = SpanF {a ∈ A | deg(a) ≤ m}M = SpanF

{ae1

1 · · · aekk | k, e1, . . . , ek ∈ N, e1 + · · · + ek < d

a1, . . . , ak ∈ N

}.

Putting the squeeze on the Noether gap 507

We are going to show thatB · M = A, i.e., thatM generatesA as aB-module.

If a ∈ A, then the minimal polynomial ofa has degree at mostd. Hencewe can findb1, . . . , bd−1 ∈ C such that

(∗) ad = −(b1ad−1 + · · · + bd) .

If deg(a) ≤ m then

deg(b1) < deg(b2) < · · · < deg(bd) = dm

sob1, . . . , bd ∈ B. The elementsa, a2, . . . , ad−1 belong toM so(∗) showsad ∈ B · M for anya ∈ N .

Next suppose thataE = ae11 · · · aek

k with a1, . . . , ak ∈ N ande1 + · · ·+ek = d. From Lemma 2.1 we obtain

(∗∗) (−1)dd !aE =∑

I⊆{1,...,d}(−1)|I|

(∑i∈I

ai

)d

=∑

I⊆{1,...,d}(−1)|I|hd

I ,

wherehI ∈ N . Sinced ! 6= 0 ∈ F it follows thataE ∈ B · M .Assume inductively that all monomialsaE = ae1

1 · · · aekk with k, e1, . . . ,

ek ∈ N, a1, . . . , ak ∈ N ande1+· · ·+ek ≤ d+i belong toB ·M . Considera monomialaE = ae1

1 · · · aekk with k, e1, . . . , ek ∈ N, a1, . . . , ak ∈ N and

e1 + · · ·+ ek = d+ i+1. Without loss of generality we may supposeaE =aE′ · ak. By the induction hypothesis we haveaE′ ∈ B · M and thereforewe may chooseh1, . . . , hl ∈ N andd1, . . . , dl ∈ N with d1 + · · · + dl < dandcD ∈ B so that

aE′=∑

cDhD =∑

|D|<d−1

cDhD +∑

|D|=d−1

cDhD ,

where

hD =l∏

i=1

hdii .

If |D| < d − 1 thenhDak ∈ M for degree reasons and hence∑|D|<d−1

cDhDak ∈ B · M .

If |D| = d − 1 then by(∗∗) hDak ∈ B · M and hence∑|D|=d−1

cDhDak ∈ B · M .

508 L. Smith

Combining these inclusions gives

aE = aE′ · ak =∑

cDhDak

=∑

|D|<d−1

cDhDak +∑

|D|=d−1

cDhDak ∈ B · M .

Therefore, by induction, any monomialaE = ae11 · · · aek

k , with a1, . . . , ak ∈N , belongs toB · M . SinceN generatesA as an algebra we have shownthatB · M = A as required.

To complete the proof thatB = C we apply the projectionπ to A andobtain

C = π(A) = π(B · M) = B

sinceπ(M) ⊂ B. utTheorem 2.3. Letρ : G ↪→ GL(n, F) be a representation of a finite groupG over the fieldF of characteristicp. If H ≤ G is a subgroup such thatp > |G : H|, then

β(ρ) ≤ β(ρ |H) · |G : H| .

Proof. Consider the inclusionF[V ]G ⊆ F[V ]H . This is a finite extension,and everyf ∈ F[V ]H is a root of the polynomial

Φf (X) =∏

g∈G/H

(X − gf) ,

where the product is taken over a set of coset representatives ofH in G. ThepolynomialΦf (X) has degree|G : H| and thereforedeg(F[V ]H |F[V ]G) ≤|G : H|. Sincep does not divide|G : H| there is the averaging operator,derived from the relative transferTrG

H ,

π =1

|G : H|∑

g∈G/H

g =1

|G : H|TrGH : F[V ]H −→ F[V ]G ,

(see [3] Section 4.2) which splits the inclusionF[V ]G ↪→ F[V ]H . Thereforethe hypotheses of Theorem 2.2 are satisfied, and, applying this theoremyields the desired conclusion.ut

As an easy consequence we obtain inductively that Noether’s boundholds for the alternating groups in the nonmodular case.

Corollary 2.4. Let An be the alternating group onn letters andF a fieldof characteristic prime to|An|, thenβF(An) ≤ |An|.

Putting the squeeze on the Noether gap 509

Proof. The alternating groupA3 is cyclic, andA4 is solvable, so the resultholds for them by [4]. Let the characteristic ofF be p, and, note thatpdoes not divide|An| = n!

2 if and only if p > n = |An : An−1|. Hence,proceeding inductively onn, and applying Theorem 2.3, we obtain

βF(An) ≤ βF(An−1) · |An : An−1| ≤ (n − 1)!2

· n = |An| ,

so the result follows. ut

Theorem 2.3 has a number of other consequences, among which we notethe following.

Corollary 2.5. Let F be a field of characteristicp, G a finite group, andH < G a subgroup. Ifp > |G : H| and βF(H) ≤ |H|, thenβF(G) ≤|G|. ut

This can be applied to some of the sporadic simple groups to almostclose the Noether gap for them. For example, looking at the entries for theMathieu groups in theATLAS [1], one can derive the following result.

Proposition 2.6. Let Mi be one of the Mathieu groups, i=10, 11, 12, 22,23, or 24, andF a field of characteristicp with p > i. ThenβF(Mi) ≤|Mi|. ut

The following little table lists the primes that do not divide the orderof a Mathieu group but are excluded by this proposition, so for which theNoether gap is not yet closed.

group primes excludedM10, M11, M12 7M22, M23, M24 17, 19

Finally, we note that Theorem 2.3 can be applied iteratively to obtain:

Corollary 2.7. Let F be a field of characteristicp, G a finite group and

{1} < Gk < Gk−1 < · · · < G1 < G0 = G

a chain of subgroups ofG such that:

(i) βF(Gi) ≤ |Gi|, for i = 1, . . . , k − 1, and(ii) p > max {|Gi : Gi−1| | i = 1, . . . , k − 1},

thenβF(G) ≤ |G|. ut

510 L. Smith

References

1. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, with computationalassistance from J.G. Thackray, Atlas of Finite Groups. Claerndon Press, Oxford, 1985

2. E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann.77(1916), 89—92

3. L. Smith, Polynomial Invariants of Finite Groups, (second printing). A.K. Peters Ltd.,Wellesley, MA 1995, 1997

4. L. Smith Noether’s Bound in the Invariant Theory of Finite Groups. Arch. der Math.66(1996), 89–92

5. L. Smith, R. E. Stong, On the Invariant Theory of Finite Groups: Orbit Polynomials andSplitting Principles. J. of Algebra110(1987), 134–157

6. H. Weyl, The Classical Groups (2nd ed.). Princeton Univ. Press, Princeton, 1946