# PUTTING THE SQUEEZE ON THE NOETHER GAP
PUTTING THE SQUEEZE ON THE NOETHER GAP
PUTTING THE SQUEEZE ON THE NOETHER GAP
PUTTING THE SQUEEZE ON THE NOETHER GAP
PUTTING THE SQUEEZE ON THE NOETHER GAP
THE CASE OF THE ALTERNATING GROUPS A n
THE CASE OF THE ALTERNATING GROUPS A n
THE CASE OF THE ALTERNATING GROUPS A n
THE CASE OF THE ALTERNATING GROUPS A n
THE CASE OF THE ALTERNATING GROUPS A n
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
# AND
SCHOOL OF MATHEMATICS
UNIVERSITY OF MINNESOTA
ORDWAY LECTURES
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : Let # : G # GL(n, IF) be a faithful representation of the finite group G over the
field IF. In 1916 E. Noether proved that for IF of characteristic zero the ring of invariants
IF[V ] G is generated as an algebra by the invariant polynomials of degree at most ïGï. This
result has been generalized to the case where the characteristic of IF is greater than ïGï, or
when the characteristic of IF is prime to the order of G and the group G is solvable. In this
note we prove that if Noether's bound fails in the nonmodular case, then it fails for a finite
nonabelian simple group. We then show how yet another reworking of Noether's argument
leads to a proof that Noether's bound holds in the nonmodular case for the alternating groups.
MATHEMATICS SUBJECT CLASSIFICATION : 13A50 Invariant Theory
Typeset by LS T E X
Let G be a finite group and # : G # GL(n, IF) a faithful representation of G over the field
IF. Via # the group G acts on the vector space V = IF n , and hence also on the graded algebra
IF[V ] of homogeneous polynomial functions on V . The subalgebra of functions IF[V ] G fixed
by the action of G is called the ring of invariants of G. As a general reference on invariant
theory see [3].
In [2] E. Noether proved that whenever the characteristic of IF is zero then IF[V ] G is a finitely
generated as an algebra over IF. In addition she gave an algorithm to construct a system of
generators using polarizations of elementary symmetric polynomials (see also [6] VIII.B.15 or
[3] §3.3). From this she deduced that IF[V ] G is generated by invariant polynomials of degree
at most ïGï. This upper bound for the degrees of the generators in a minimal generating set
is referred to as Noether's bound.
If the ground field IF has characteristic p, then (see [5] or [3] Theorem 3.1.10) Noether's bound
is known to hold if p > ïGï, where ïGï denotes the order of the group G. It may fail (see for
example [3] §4.2 Example 2) if p divides ïGï. If p does not divide ïGï, but, p < ïGï, then
Noether's bound is known to hold for solvable groups [4], and it has been conjectured to hold
in general. The region, p < ïGï, but p # ïGï, where Noether's bound is not known either to
hold or to fail, is the Noether gap of the title. The first goal of this note is to show that
if Noether's bound fails in the nonmodular case, then it fails for a finite nonabelian simple
group. We then show how yet another reworking of one of Noether's argument leads to a proof
for the alternating groups that Noether's bound holds in the nonmodular case.
This research was done while preparing a lecture on Noether's bound for a Crash Course on
Invariant Theory given by the author as Ordway Visitor at the School of Mathematics of the
University of Minnesota. I would like to thank the more than twenty participants in this
lecture series for their attentive and critical attitude, and probing questions.
§1. Reducing the Noether Gap to Simple Groups
Let us begin by introducing some notation. If A is a graded connected commutative
Noetherean algebra over a field IF, denote by b(A) the maximum degree of a (homoge
neous) generator of A in any minimal generating set for A as an IF algebra. This is noth
ing but the degree of the Poincar’ e polynomial P(QA, t) of the module of indecomposables
QA := IF Ä A
•
A = •
A / ( •
A) 2 , where •
A # A denotes the augmentation ideal. (See [3] Chapter 4
and Theorem 5.2.5.) Hence b(A) does not depend on the choice of minimal generating set. If
G is a finite group and # : G # GL(n, IF) a faithful representation of G over the field IF,
then we denote b(IF[V ] G ) by b(#). We set
b IF (G) = max n b(#)## : G # GL(n, IF) o .
The following proposition is proved by a simple linearization process as in [4].
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1 : Let A be a graded connected commutative algebra over a field IF and
G £ Aut * (A) a finite group of grading preserving automorphisms of A. If ïGï ÎIF × then
b(A G ) £ b(A)b IF (G).
LARRY SMITH
PROOF PROOF PROOF PROOF PROOF : Let V =
b(A)
Å
i=1
A i , where A i denotes the homogeneous component of A of degree i.
The group G acts on V by linear automorphisms, and the natural map
S(V ) ## A
is a Gequivariant epimorphism, where S(V ) denotes the symmetric algebra on the graded
vector space A. (Apart from the grading on V this is just IF[V * ], where V * is the dual of V .)
Since the characteristic of IF does not divide the order of G the induced map
S(V ) G
# A G
is also onto, and, taking account of the maximum degree of an element of V , yields the result.
COROLLARY 1.2
COROLLARY 1.2
COROLLARY 1.2
COROLLARY 1.2
COROLLARY 1.2 : Let G be a finite group, N # G a normal subgroup, and IF a field of
characteristic p. Suppose # : G # GL(n, IF) is a representation of G over IF and ## N the
restriction of # to N . Then G / N acts on IF[V ] N and, if p # ïGï, then
b(#) £ b(## N )b IF (G / N) ,
and hence
b IF (G) £ b IF (N)b IF (G / N).
PROOF PROOF PROOF PROOF PROOF : This follows from the isomorphism (see [3] Proposition 1.5.1)
IF[V ] G ~
= IF[V ] N
G
and Proposition 1.1.
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3 : Let G be a finite group and IF a field. Suppose the characteristic of
IF does not divide the order of G. If G has a composition series whose composition factors
K 1 , . . . , Km satisfy b IF (K i ) £ ïK i ï, for i = 1 , . . . , m, then b IF (G) £ ïGï.
PROOF PROOF PROOF PROOF PROOF : This follows from Corollary 1.2 by induction on m.
COROLLARY 1.4
COROLLARY 1.4
COROLLARY 1.4
COROLLARY 1.4
COROLLARY 1.4 : Let IF be a field of characteristic p. If there is a finite group G with
p # ïGï for which b IF (G) > ïGï, then there is also a finite nonabelian simple group S with
b IF (S) > ïSï.
PROOF PROOF PROOF PROOF PROOF : Choose G of minimal order such that p # ïGï and b IF (G) > ïGï. Then G is simple.
For, if not, we may choose a nontrivial normal subgroup N # G. Since ïNï, ïG / Nï < ïGï it
follows that b IF (N) < ïNï and b IF (G / N) < ïG / Nï, whence from Corollary 1.2 we obtain
b IF (G) £ b IF (N )b IF (G / N ) £ ïNï · ïG / Nï = ïGï ,
which would be a contradiction. Hence G is simple, and by [4] Lemma 1 it is not cyclic of
prime order, so must be nonabelian.
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PUTTING THE SQUEEZE ON THE NOETHER GAP
§2. Closing the Noether Gap for the Alternating Groups
We begin with a generalization of [3] Theorem 2.4.2. For this we require some terminol
ogy. Suppose that C # A is a finite integral extension of graded connected commutative
Noetherean algebras over a field. If a ÎA then a is a root of a monic polynomial of minimal
degree m a (X ) ÎC[X ], called the minimal polynomial of a over C. We set
deg(AïC) = max n deg(m a (X ))#a ÎA o
and call it the degree of A over C. As usual, if there is no maximum, we set deg(AïC) = #.
Recall the following combinatorial lemma from [3] Lemma 2.4.1.
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1 : Let V be a vector space over a field IF and u 1 , . . . , u j ÎIF[V ]. If j! #= 0 ÎIF
then the monomial u 1 · · · u j is a linear combination of jth powers of sums of elements of
{u 1 , . . . , u j }.
PROOF PROOF PROOF PROOF PROOF : This follows from the formula
(-1) j j ! u 1 · · · u j = X
I#{1 ,..., j}
(-1) ïIï 0 @ X iÎI
u i
1 A
j
.
In this formula, I runs over all subsets of {1 , . . . , j} and ïIï is the cardinality of I.
THEOREM 2.2
THEOREM 2.2
THEOREM 2.2
THEOREM 2.2
THEOREM 2.2: Let C # A be a finite integral extension of graded connected commutative
Noetherean algebras over the field IF of characteristic p. Suppose
(i) deg(AïC) is finite,
(ii) p > deg(AïC), and
(iii) there exists a Cmodule splitting
p : A # C
to the inclusion C # A.
Then b(C) £ b(A) · deg(AïC).
PROOF PROOF PROOF PROOF 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 b(A) = m and deg(AïC) = d. Let B be the subalgebra of C generated by elements of degree
at most md. Our goal is to show that B = C. To this end introduce
N = Span IF n a ÎA ï deg(a) £ m o
M = Span IF a e 1
1 · · · a e k
k # k, e 1 , . . . , e k ÎIN, e 1 + · · · + e k < d
a 1 , . . . , a k ÎN .
We are going to show that B · M = A, i.e., that M generates A as a Bmodule.
If a ÎA, then the minimal polynomial of a has degree at most d. Hence we can find
b 1 , . . . , b d-1 ÎC such that
(*) a d = -(b 1 a d-1 + · · · + b d ).
If deg(a) £ m then
deg(b 1 ) £ deg(b 2 ) £ · · · £ deg(b d ) £ dm
so b 1 , . . . , b d ÎB. The elements a, a 2 , . . . , a d-1 belong to M so (*) shows a d ÎB · M for any
a ÎN .
3
LARRY SMITH
Next suppose that a E = a e 1
1 · · · a e k
k with a 1 , . . . , a k ÎN and e 1 + · · · + e k = d. From lemma 2.1
we obtain
(**) (-1) d d !a E = X
I#{1 ,..., d}
(-1) ïIï 0 @ X iÎI
a i
1 A
d
= X
I#{1 ,..., d}
(-1) ïIï h d
I ,
where h I ÎN . Since d ! #= 0 ÎIF it follows that a E ÎB · M .
Assume inductively that all monomials a E = a e 1
1 · · · a e k
k with k, e 1 , . . . , e k ÎIN, a 1 , . . . , a k ÎN
and e 1 + · · · + e k £ d + i belong to B · M . Consider a monomial a E = a e 1
1 · · · a e k
k with k, e 1 , . . . , e k Î
IN, a 1 , . . . , a k ÎN and e 1 + · · · + e k = d + i + 1. Without loss of generality we may suppose
a E = a E # · a k . By the induction hypothesis we have a E # ÎB · M and therefore we may choose
h 1 , . . . , h l ÎN and d 1 , . . . , d l ÎIN with d 1 + · · · + d l < d and c D ÎB so that
a E # = X c D h D = X
ïDï ïG : Hï, then
b(#) £ b(## H ) · ïG : Hï.
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PUTTING THE SQUEEZE ON THE NOETHER GAP
PROOF PROOF PROOF PROOF PROOF : Consider the inclusion IF[V ] G
# IF[V ] H . This is a finite extension, and every
f ÎIF[V ] H is a root of the polynomial
F f (X ) = Y
gÎG / H
(X - g f ) ,
where the product is taken over a set of coset representatives of H in G. The polynomial
F f (X ) has degree ïG : Hï and therefore deg(IF[V ] H ïIF[V ] G ) £ ïG : Hï. Since p does not
divide ïG : Hï there is the averaging operator, derived from the relative transfer Tr G
H ,
p = 1
ïG : Hï X
gÎG / H
g = 1
ïG : Hï
Tr G
H : IF[V ] H
# IF[V ] G ,
(see [3] Section 4.2) which splits the inclusion IF[V ] G
# IF[V ] H . Therefore the hypotheses
of Theorem 2.2 are satisfied, and, applying this theorem yields the desired conclusion.
As an easy consequence we obtain inductively that Noether's bound holds for the alternating
groups in the nonmodular case.
COROLLARY 2.4
COROLLARY 2.4
COROLLARY 2.4
COROLLARY 2.4
COROLLARY 2.4: Let A n be the alternating group on n letters and IF a field of characteristic
prime to ïA n ï, then b IF (A n ) £ ïA n ï.
PROOF PROOF PROOF PROOF PROOF : The alternating group A 3 is cyclic, and A 4 is solvable, so the result holds for them
by [4]. Let the characteristic of IF be p, and, note that p does not divide ïA n ï = n!
2
if and
only if p > n = ïA n : A n-1 ï. Hence, proceeding inductively on n, and applying Theorem 2.3,
we obtain
b IF (A n ) £ b IF (A n-1 ) · ïA n : A n-1 ï £ (n - 1)!
2
· n = ïA n ï
and the result follows.
Theorem 2.3 has a number of other consequences, among which we note the following.
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5 : Let IF be a field of characteristic p, G a finite group, and H < G a
subgroup. If p > ïG : Hï and b IF (H) £ ïHï, then b IF (G) £ ïGï.
This can be applied to some of the sporadic simple groups to almost close the Noether gap for
them. For example, looking at the entries for the Mathieu groups in the ATLAS [1], one can
derive the following result.
PROPOSITION 2.6
PROPOSITION 2.6
PROPOSITION 2.6
PROPOSITION 2.6
PROPOSITION 2.6 : Let M i be one of the Mathieu groups, i=10, 11, 12, 22, 23, or 24, and IF
a field of characteristic p with p > i. Then b IF (M i ) £ ïM i ï.
The following little table lists the primes that do not divide the order of a Mathieu group but
are excluded by this proposition, so for which the Noether gap is not yet closed.
GROUP PRIMES EXCLUDED
M 10 , M 11 , M 12 7
M 22 , M 23 , M 24 17, 19
Finally, we note that Theorem 2.3 can be applied iteratively to obtain:
5
LARRY SMITH
COROLLARY 2.7
COROLLARY 2.7
COROLLARY 2.7
COROLLARY 2.7
COROLLARY 2.7 : Let IF be a field of characteristic p, G a finite group and
1
< G k < G k-1 < · · · < G 1 < G 0 = G
a chain of subgroups of G such that:
(i) b IF (G i ) £ ïG i ï, for i = 1 , . . . , k - 1, and
(ii) p > max n ïG i : G i-1 ï#i = 1 , . . . , k - 1 o ,
then b IF (G) £ ïGï.
References
[1] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, with computational
assistance 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 and R. E. Stong, On the Invariant Theory of Finite Groups: Orbit Polynomials
and Splitting Principles, J. of Algebra 110 (1987), 134--157.
[6] H. Weyl, The Classical Groups (second edition), Princeton Univ. Press, Princeton, 1946.
Larry Smith
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
and
# Mathematisches Institut der Universit ˜
at
D 37073 G˜ ottingen
Federal Republic of Germany
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