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On \$\cal A\$-generators for the cohomology of the

symmetric and the alternating groups

NGUYEN H V HUNG

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 139 / Issue 03 / November 2005,

pp 457 - 467

DOI: 10.1017/S0305004105008674, Published online: 21 October 2005

Link to this article: http://journals.cambridge.org/abstract_S0305004105008674

How to cite this article:

NGUYEN H V HUNG (2005) On $\cal A$-generators for the cohomology of the symmetric and the alternating groups Mathematical Proceedings of the Cambridge Philosophical Society, 139, pp 457-467 doi:10.1017/S0305004105008674

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Math Proc Camb Phil Soc (2005), 139, 457
2005 Cambridge Philosophical Societyc

doi:10.1017/S0305004105008674 Printed in the United Kingdom

457

On A-generators for the cohomology of the symmetric

and the alternating groups

ByNGUY ˆE˜ N H V HU.NG†

Department of Mathematics, Vietnam National University,

334 Nguyˆe˜n Tr˜ai Street, Hanoi, Vietnam.

(Received 23 March 2004)

Abstract

Following Quillen’s programme, one can read off a lot of information on the

co-homology of a finite group G by studying the restriction homomorphism from this cohomology to the cohomology of all maximal elementary abelian subgroups of G.

This leads to a natural question on how much information onA-generators of H ∗ (G)

one can read off from using the restriction homomorphism, where A denotes the

Steenrod algebra In this paper, we show that the restriction homomorphism gives,

in some sense, very little information onA-generators of H ∗ (G) at least in the im-portant three cases, where G is either the symmetric group, the alternating group,

or a certain type of iterated wreath products

1 Introduction and statement of results Throughout this paper, by the cohomology H ∗ (G) of a finite group G we mean the mod 2 cohomology H ∗ (BG;F2) of its classifying space BG.

Following Quillen [13], one of the most important step in the study of the cohomo-logy of a finite group G is to investigate the restriction homomorphism

A ∈E(G)

H ∗ (A),

where the product runs over E(G), the set of all maximal elementary abelian 2-subgroups A of G Indeed, Quillen shows that every element in the kernel of Res G is

nilpotent In particular, in the interesting case where G is the symmetric group Σ m

of all permutations on m letters, he proves that the restriction homomorphism

A∈E(Σ m)

H ∗ (A)

is a monomorphism

cohomo-logy of a finite group G From Quillen’s point of view, it is natural to ask how much

one can see the minimal A-generators of H ∗ (G) by studying that of H ∗ (A), where

† The work was supported in part by the National Research Program, Grant N ◦140804.

458 Nguy ˆ e ˜ n H V Hung

A denotes a maximal elementary abelian 2-subgroup of G More precisely, how can

one read offF2⊗

∗ (G) from the homomorphism

F2⊗

∗ (G) → F2⊗

∗ (A) induced by the inclusion A ⊂ G? Here A acts upon F2via the augmentationA → F2

The aim of this paper is to study this problem in three important cases; G is either the

symmetric group the alternating group or a certain type of iterated wreath products From Quillen’s point of view, the following theorems and proposition are somewhat unexpected on minimalA-generators for the cohomology of finite groups.

Let Σmbe thought of as the symmetric group ΣX on a set X of cardinality |X| = m Let A be a maximal elementary abelian 2-subgroup of Σ m and O ⊂ X an orbit of the group A As A is a 2-group, the cardinality of O is a power of 2 Denote by A | O

the group of all the restrictions g | O for g ∈ A.

Theorem 1·1 Let A be a maximal elementary abelian 2-subgroup of Σ m and O an orbit of A with cardinality |O| > 4 Then, the homomorphism

Res :F2⊗

∗(Σm)→ F2⊗

∗ (A | O)

induced by the inclusion A | O ⊂ A ⊂ Σ m is trivial in positive degrees.

LetAm be the alternating group on m letters.

Theorem 1·2 Let B be a maximal elementary abelian 2-subgroup of A m and O an orbit of B with cardinality |O| > 4 Then, B| O ⊂ A |O | and the homomorphism

Res :F2⊗

∗(Am)→ F2⊗

∗ (B | O)

induced by the inclusion B | O ⊂ B ⊂ A m is trivial in positive degrees.

LetVk denote an elementary abelian 2-group of rank k Suppose G is either Σ2n

orA2n and E
Z/2 Then, the regular permutation representation V n ⊂ G induces

a natural inclusionVk
Vn × E k −n ⊂ G  k −n E for k
n (See Section 4 for details.)

Proposition 1·3 Let G be either Σ2n orA2n and E
Z/2, then the homomorphism

Res :F2⊗

∗ (G  k −n E) → F2⊗

∗(Vk)

induced by the inclusionVk ⊂ G  k −n E is trivial in positive degrees for k
n > 2.

The two theorems and the proposition are expositions of an algebraic version of

the classical conjecture on spherical classes (See [1–5, 14].) The theorems fail for

|O| = 2 or 4 and so does the proposition for n  2 because of the existence of the

Hopf invariant one and the Kervaire invariant one classes

In the proofs of Theorems 1·1, 1·2 and Proposition 1·3, one will see that: if G is

either the symmetric group, the alternating one, or a certain type of iterated wreath products, andVk is some elementary abelian 2-subgroup of G, then A-generators for

H ∗ (G) cannot be read off from A-generators for the polynomial algebra H ∗(Vk)

P k  F2[x1, , x k ] with deg (x i) = 1 However, they could be read off from A-generators of the invariant algebra P W , for some subgroups W of GL(k,F2) This is a

Cohomology of the symmetric and the alternating groups 459 motivation for the study ofA-generators of P W

k , where W is a subgroup of GL(k,F2)

(See [8, 9] for such a study and its application in the case of W = GL(k,F2).) The paper contains four sections The case of the symmetric groups and that of the alternating groups are respectively studied in Sections 2 and Section 3 Finally, the case of the wreath products is investigated in Section 4

2 The case of the symmetric groups Let A be a maximal elementary abelian 2-subgroup of Σ m and O1, O2, , O t all

the orbits of A Then we have an inclusion

A ⊂ A| O1× A| O2× · · · × A| O t

As A is an elementary abelian 2-group, then so is A | O i for every i Further, since A

is a maximal elementary abelian 2-subgroup of Σm, we get the equality

A = A| O1× A| O2× · · · × A| O t The following lemma is obvious (See e g [10].)

Lemma 2·1 Suppose C is an elementary abelian 2-group, which acts faithfully and transitively on a set O Then, there is anF2-vector space structure V on O such that C is isomorphic to (the group of all translations on) the additive group V.

Let 0 be a fixed point of O For any g ∈ C, we set v g = g(0) As C acts faithfully and transitively on O, we get O = {v g | g ∈ C} Suppose C is an additive group Obviously, O is equipped with anF2-vector space structure, denoted byV, by setting

v g + v h = v g +h ,

av g = v ag , for g, h ∈ C, a ∈ F2

For every v ∈ V, there is h ∈ C such that v = h(0) = v h Then, for any g ∈ C, we

have

g(v) = g(h(0)) = (g + h)(0) = v g +h = v g + v h = v + v g Thus, g is the translation by v g onV So, C is a subgroup of (the group of all

trans-lations on) the additive group V Further, since the abelian group C acts faithfully

and transitively onV, we get |C| = |V| In conclusion, C
V.

As O i is an orbit of A, it is easily seen that A | O i acts faithfully and transitively on

the orbit O ifor 1 i  t Therefore, by the lemma, we have

A | O i
Vk i

for 1 i  t, where V k i is a vector space of certain dimension k ioverF2

According to M `ui [10], the group

A = A| O1× A| O2× · · · × A| O t
Vk1× · · · × V k t

is a maximal elementary abelian 2-subgroup of Σm = ΣX if and only if there is at

most one of the orbits O1, , O twith cardinality 1, or equivalently there is at most

one of the dimensions k equaling 0

460 Nguy ˆ e ˜ n H V Hung

To prepare for the proof of Theorem 1·1, we first consider the case where A has exactly one orbit O = X Then, A = A | O is isomorphic to the additive group of a

k-dimensionalF2-vector spaceV = Vk with|V| = |O| = |X| = 2 k So, the symmetric group ΣX is isomorphic to Σ2k

Proposition 2·2 Let V = V k be an elementary abelian 2-group of rank k > 2 Then, the homomorphism

Res :F2⊗

∗(Σ2k)−→ F2⊗

∗(V)

induced by the regular permutation representation V ⊂ Σ2k is trivial in positive degrees Proof. It is well known that the Weyl group ofV in Σ2k is the general linear group

GL( V) = GL(k, F2) Further, according to M `ui [10], the image of the restriction

homomorphism

Res : H ∗(Σ2k)−→ H ∗(V)

is nothing but the Dickson algebra H ∗(V)G L (V) consisting of all invariants in H ∗(V)

under the regular action of GL(V).

Therefore, the homomorphism Res : F2⊗

∗(Σ2k) → F2⊗

∗(V) factors through

F2⊗

A (H ∗(V)G L (V)) More precisely, we have a commutative diagram

F2⊗

∗(V)

R es

F2⊗

A (H ∗(V)G L (V)) ,

@

@

@

@



where the homomorphismF2⊗

A (H ∗(V)G L (V))→ F2⊗

∗(V) is induced by the inclusion

H ∗(V)G L (V) ⊂ H ∗(V) This map is shown by Hu.ng – Nam [6] to be zero in positive

degrees for k > 2.

As a consequence, the map Res : F2⊗

∗(Σ2k) → F2⊗

∗(V) is zero in positive

degrees for k > 2.

A (H ∗(V)G L (V))→ F2⊗

∗(V) is zero in positive

de-grees for k > 2 is equivalent to the fact that the Dickson algebra H ∗(V)G L (V) with dim (V) > 2 is a subset of the image of the action of the maximal ideal in the

Steen-rod algebra on the polynomial algebra H ∗(V) This is an exposition of an algebraic

version of the classical conjecture on spherical classes It fails for k = dim(V) = 1 or

2 because of the existence of respectively the Hopf invariant one and the Kervaire

invariant one classes (See [2] for details.)

Proof of Theorem1·1 From the hypothesis, O is one of the orbits O1, , O t and

A | O is one of the summands in the decomposition

A = A| O1× A| O2× · · · × A| O t Obviously, A | O acts faithfully and transitively on O As A is a 2-group, the cardinality

of O is a power of 2 If |O| = 2 k, then by Lemma 2·1 A| is isomorphic to the additive

Cohomology of the symmetric and the alternating groups 461

group of a k-dimensional F2-vector space V = Vk and ΣO = Σ2k is a subgroup of

ΣX = Σm We get the inclusions of groups

A | O =V ⊂ Σ2k ⊂ Σ m

The inclusions show that the homomorphism

Res :F2⊗

∗(Σm)→ F2⊗

∗ (A | O) factors throughF2⊗

∗(Σ2k) That is, we have a commutative diagram

F2⊗

∗ (A | O)

R es

F2⊗

∗(Σ2k)

@

@

@

@



By Proposition 2·2, the map

Res :F2⊗

∗(Σ2k)→ F2⊗

∗ (A | O)

is trivial in positive degrees for k > 2, or equivalently for |O| = 2 k > 4 The theorem

follows

3 The case of the alternating groups

Lemmas 3·1 and 3·2 deal with some results by H M`ui in [10] and [11] on elementary

abelian 2-subgroups of the alternating groups To make the paper self-contained, we will re-express his proofs for these lemmas The following two lemmas are obvious consequences of the first two A complete classification of all maximal elementary

abelian 2-subgroups of the alternating groups up to conjugacy is given in [12].

However, we will not use this classification in this paper

Suppose again thatVkis an elementary abelian 2-group andVk ⊂ Σ2k is its regular permutation representation In other words,Vk acts on itself by translation, while

Σ2k is thought of as the symmetric group on (the point set of)Vk So, the alternating subgroupA2k also acts onVk

Lemma 3·1 V k ⊂ A2k for k
2.

Proof Each element g ∈ V k, regarded as a translation onVk, is of order 2 Thus,

(x, g(x)) is a product of 2 k −1 transpositions If k
2, then 2k −1 is even So, g

is an even permutation The lemma follows

The general linear group GL k = GL(V k) acts regularly onVk

Lemma 3·2 GL k ⊂ A2k for k > 2.

Proof Choose a basis (e1, e2, , e k) ofVk Then, GL k can be identified with the

group of all invertible k × k-matrices with entries in F2 Let B ijbe the matrix, whose entries are zero except the ones on the main diagonal and the one appearing in the

462 Nguy ˆ e ˜ n H V Hung

{B ij | 1  i  j  k} Indeed, the multiplication of B ij from the left to a matrix B is the addition of the jth row to the ith row of B; while the multiplication of B ij from

the right to a matrix B is the addition of the ith column to the jth column of B Each matrix B ∈ GL k can be transformed into the identity matrix by multiplications with

some matrices B ij either from the left or from the right

Since i  j, we have B ij (x) = x if and only if x belongs to the (k − 1)-dimensional vector space Span (e1, , ˆe j , , e k ) As B ij is of order 2, it is a product of 2k −1 /2 =

2k−2 transpositions Therefore, if k > 2, then B ij is an even permutation Hence,

GL k ⊂ A2k for k > 2.

Suppose H is a subgroup of G Let N G (H), C G (H), W G (H) denote respectively the normalizer, the centralizer and the Weyl group of H in G.

Lemma 3·3 WA2k(Vk ) = GL k for k > 2.

Proof As is well known, NΣ2k(Vk) = Vk ×GL k and CΣ2k(Vk) = Vk (see e.g [10]).

Using Lemmas 3·1 and 3·2, we have

NA

2k(Vk ) = NΣ2k(Vk) A2k = (Vk ×GL k) A2k =Vk ×GL k ,

CA

2k(Vk ) = CΣ2k(Vk) A2k =Vk  A2k =Vk

for k > 2 So, we get

2k(Vk ) = NA2k(Vk )/CA2k(Vk) =Vk ×GL k /Vk = GL k

for k > 2.

Lemma 3·4 Let B be a maximal elementary abelian 2-subgroup of A n If O is an orbit

of B with cardinality |O| = 2 k
4, then B| O ⊂ A |O | Further, there exists anF2-vector space structureVk on O such that B | O is isomorphic to (the group of all translations on) the additive group ofVk

Proof As O is an orbit of B, it is easy to see that B | Oacts faithfully and transitively

on O Then, by Lemma 2 ·1, there exists an F2-vector space structureVk on O such that B | O is isomorphic to (the group of all translations on) the additive group ofVk

On the other hand, by Lemma 3·1, V k ⊂ A2k =A|O | for k
2

Proposition 3·5 Let V = V k be an elementary abelian 2-group of rank k > 2 Then, the homomorphism

Res :F2⊗

∗(A2k)→ F2⊗

∗(V)

induced by the inclusion V ⊂ A2k is trivial in positive degrees.

Proof. The proof is similar to that of Proposition 2·2 By Lemma 3·3, WA2k(Vk) =

GL k for k > 2 Hence, the restriction homomorphism Res : H ∗(A2k)→ H ∗(V) factors

through the Dickson algebra H ∗(V)G L (V) So, the induced homomorphism

Res :F2⊗

∗(A2k)→ F2⊗

∗(V) factors through F2⊗

A (H ∗(V)G L (V)) The proposition follows from the fact that the homomorphism

F2⊗ A



H ∗(V)G L (V)

∗(V)

Cohomology of the symmetric and the alternating groups 463

induced by the inclusion H ∗(V)G L (V) ⊂ H ∗(V) equals zero in positive degrees for

k > 2 (See Hu.ng–Nam [6].)

The proof is complete

2-group, the cardinality of O is a power of 2 From the hypothesis, |O| = 2 k > 4, thus k > 2 Then, by Lemmas 3 ·1 and 3·4, we get

B | O
V = Vk ⊂ A2k ⊂ A m

The homomorphism

Res :F2⊗

∗(Am)→ F2⊗

∗ (B | O)
F2⊗

∗(V)

factors through Res : F2⊗

∗(A2k) → F2⊗

∗(V) By Proposition 3·5, the last

homomorphism is trivial in positive degrees for k > 2.

4 On the case of iterated wreath products Suppose G is a finite group and E
Z/2 Let G  E  (G × G) ×E be the wreath product of G by E, where E acts on G × G by permutation of the factors The group

G  k E is defined by induction on k as follows

G 1E = G  E, G  k E =



G  k −1 E

 E.

One regards G × E as a subgroup of G  E via the inclusion

(g, a) −→ (g, g; a), for g ∈ G, a ∈ E So, by induction on k, G × E k is a subgroup of G  k E.

Suppose H is a subgroup of GL n and K is a subgroup of GL k−n for k
n We are

interested in the following group



 GL k ,

where∗ denotes any n × (k − n) matrix with entries in F2 In particular, we focus

on the important two cases, where K is either the unit subgroup 1 k −n or the Sylow

2-subgroup T k −n of GL k −n consisting of all upper triangular matrices with 1 on the main diagonal

Lemma 4·1 Let V n  G  Σ2n , where the inclusionVn ⊂ Σ2n is the regular permuta-tion representapermuta-tion ofVn Then the Weyl group W G(Vn ) is isomorphic to a subgroup of

GL n
GL(V n ) and

W G E (V n × E)
W G(Vn)• 11  GL n +1 Proof By definition, σ ∈ C G(Vn ) if and only if σtσ −1 = t for every t ∈ V n In particular, we have

σt(0) = tσ(0), σ(t(0)) = σ(0) + t(0).

464 Nguy ˆ e ˜ n H V Hung

AsVn acts transitively on itself by translation, each element u ∈ V n can be written

in the form u = t(0), for some t ∈ V n Hence, σ is nothing but the translation by σ(0)

onVn Therefore, C G(Vn)⊂ V n On the other hand, asVn is abelian,Vn ⊂ C G(Vn)

In summary, we get C G(Vn) =Vn By similarity, C GE(Vn × E) = V n × E.

Let us consider the conjugacy homomorphism

j = j G : N G(Vn)→ Aut(V n)
GL n ,

g → (c g : v → gvg −1 ).

By definition of the centralizer subgroup, Ker(j G ) = C G(Vn) =Vn Thus

W G(Vn ) = N G(Vn )/C G(Vn ) = N G(Vn )/Ker(j G)
Im(j G ).

Therefore, W G(Vn ) is isomorphic to a subgroup of GL n Similarly, we get

W G E(Vn × E) = N G E(Vn × E)/Ker(j G E)
Im(j G E ).

So, in order to determine W GE(Vn × E), we need to compute the image of j GE Let us divide the argument into two steps

Step 1: we find the conditions for σ = (g, h; 0) ∈ N G E(Vn × E), where g, h ∈ G.

An element of V n ×E, regarded as a subgroup of GE, is either (v, v; 0) or (v, v; 1), where v ∈ V n and 0, 1 ∈ E We have

σ(v, v; 0)σ −1 = (g, h; 0)(v, v; 0)(g, h; 0) −1 = (gvg −1 , hvh −1 ; 0).

This element belongs toVn × E if and only if

gvg −1 ∈ V n , hvh −1 ∈ V n , gvg −1 = hvh −1 , for every v ∈ V n The first two conditions are respectively equivalent to g ∈ N G(Vn)

and h ∈ N G(Vn) The last condition is equivalent to

v = g −1 hvh −1 g = (g −1 h)v(g −1 h) −1 , for every v ∈ V n That is g −1 h ∈ V n = C G(Vn)

Notice that if g ∈ N G(Vn ) and g −1 h ∈ V n , then g(g −1 h)g −1 = hg −1 ∈ V n Hence

(hg −1)−1 = gh −1 ∈ V n

We now show that under the hypotheses g, h ∈ N G(Vn ) and g −1 h ∈ V n, we get

σ(v, v; 1)σ −1 = (g, h; 0)(v, v; 1)(g, h; 0) −1 = (gvh −1 , hvg −1; 1)∈ V n × E, for every v ∈ V n Indeed, this is equivalent to

gvh −1 ∈ V n , hvg −1 ∈ V n , gvh −1 = hvg −1 , for every v ∈ V n The first condition satisfies because of

gvh −1 = (gvg −1 )(gh −1)

and of (gvg −1)∈ V n , gh −1 ∈ V n Similarly, the second condition also satisfies Since

g −1 h, v ∈ V n andVn is an elementary abelian 2-group, we have

g −1 hvg −1 h = (g −1 h)2v = v.

This is equivalent to the third condition

In summary, we have shown that σ = (g, h; 0) ∈ N G E(Vn × E) if and only if

g, h ∈ N (Vn ) and g −1 h ∈ V n

Cohomology of the symmetric and the alternating groups 465 Note thatVn is an additive group, while G is a multiplicative one More precisely,

Vn is regarded as a subgroup of G via a group monomorphism

i: (Vn , +) −→ (G, ·) ⊂

So,Vn is identified with its image under i We also use the same notation to denote the induced inclusion i :Vn ×E → GE If g ∈ N ⊂ G(Vn ), then by j G (g) = A ∈ GL n

Aut(Vn) we mean

i −1 (gvg −1 ) = Ai −1 (v), for every v ∈ V n

For σ = (g, h; 0) ∈ N G E(Vn × E), we set A = j G (g) ∈ GL n and b = i −1 (gh −1)∈ V n

As gvg −1 , gh −1 ∈ V n, we obtain

i −1 (gvh −1 ) = i −1 [(gvg −1 )(gh −1 )] = i −1 (gvg −1 ) + i −1 (gh −1)

= Ai −1 (v) + b.

So, for σ = (g, h; 0) ∈ N G E(Vn × E), we get

i −1 (σ(v, v; 0)σ −1 ) = i −1 (gvg −1 , hvh −1 ; 0) = (Ai −1 (v), Ai −1 (v); 0),

i −1 (σ(v, v; 1)σ −1 ) = i −1 (gvh −1 , hvg −1 ; 1) = (Ai −1 (v) + b, Ai −1 (v) + b; 1).

Hence

j GE (σ) = j G E (g, h; 0) =

0 1 ∈ GL n +1

Step 2: similarly, σ = (g, h; 1) ∈ N G E(Vn × E) if and only if g, h ∈ N G(Vn) and

g −1 h ∈ V n

Then, setting A = j G (g) ∈ GL n and b = i −1 (gh −1)∈ V n, we have

i −1 (σ(v, v; 0)σ −1 ) = i −1 (gvg −1 , hvh −1 ; 0) = (Ai −1 (v), Ai −1 (v); 0),

i −1 (σ(v, v; 1)σ −1 ) = i −1 (gvh −1 , hvg −1 ; 1) = (Ai −1 (v) + b, Ai −1 (v) + b; 1).

So,

j G E (σ) = j G E (g, h; 1) =

0 1 ∈ GL n +1 Conversely, given A ∈ Im(j G ) and b ∈ V n , there is g ∈ N G(Vn ) such that j G (g) = A Define h by the equality b = i −1 (gh −1 ), we easily show that h ∈ N G(Vn) and that

j G E (g, h; 0) = j GE (g, h; 1) =

In conclusion, we have shown that

Im(j G E) =



0 1 A ∈ Im(j G)
W G(Vn ), b ∈ V n

.

In other words,

W GE(Vn × E)
Im(j G E)
W G(Vn)• 11.

The lemma is proved

in the form u = t(0), for some t ∈ V n Hence, σ is nothing but the translation by σ(0)

onVn Therefore, C G(Vn)⊂... ) and g −1 h ∈ V n

Cohomology of the symmetric and the. .. −1 , for every v ∈ V n The first condition satisfies because of

gvh −1 = (gvg −1 )(gh −1)

and of (gvg −1)∈