DSpace at VNU: On A-generators for the cohomology of the symmetric and the alternating groups

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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: />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

c 2005 Cambridge Philosophical Society

457

doi:10.1017/S0305004105008674 Printed in the United Kingdom

On A-generators for the cohomology of the symmetric
and the alternating groups
˜ˆ N H. V. HU.NG†
By NGUYE
Department of Mathematics, Vietnam National University,
334 Nguyˆ˜e n Tr˜ai Street, Hanoi, Vietnam.
e-mail:
(Received 23 March 2004)
Abstract
Following Quillen’s programme, one can read off a lot of information on the cohomology 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 on A-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 on A-generators of H ∗ (G) at least in the important 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 cohomology of a finite group G is to investigate the restriction homomorphism
ResG : H ∗ (G) →

H ∗ (A),
A ∈E(G )

where the product runs over E(G), the set of all maximal elementary abelian 2subgroups A of G. Indeed, Quillen shows that every element in the kernel of ResG is
nilpotent. In particular, in the interesting case where G is the symmetric group Σm
of all permutations on m letters, he proves that the restriction homomorphism
ResΣm : H ∗ (Σm ) →

H ∗ (A)
A ∈E(Σm )

is a monomorphism.
Let A be the mod 2 Steenrod algebra, which acts in the usual way on the cohomology 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

Nguye˜ˆ n H. V. Hung

A denotes a maximal elementary abelian 2-subgroup of G. More precisely, how can
one read off F2 ⊗ H ∗ (G) from the homomorphism
A

F2 ⊗ H ∗ (G) → F2 ⊗ H ∗ (A)
A

A

induced by the inclusion A ⊂ G? Here A acts upon F2 via the augmentation A → 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 minimal A-generators for the cohomology of finite groups.
Let Σm be 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 Σm 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 Σm and O an
orbit of A with cardinality |O| > 4. Then, the homomorphism
Res : F2 ⊗ H ∗ (Σm ) → F2 ⊗ H ∗ (A|O )
A

A

induced by the inclusion A|O ⊂ A ⊂ Σm is trivial in positive degrees.
Let Am be the alternating group on m letters.
Theorem 1·2. Let B be a maximal elementary abelian 2-subgroup of Am and O an
orbit of B with cardinality |O| > 4. Then, B|O ⊂ A|O | and the homomorphism
Res : F2 ⊗ H ∗ (Am ) → F2 ⊗ H ∗ (B|O )
A

A

induced by the inclusion B|O ⊂ B ⊂ Am is trivial in positive degrees.
Let Vk denote an elementary abelian 2-group of rank k. Suppose G is either Σ2n
or A2n and E % Z/2. Then, the regular permutation representation Vn ⊂ G induces
a natural inclusion Vk % Vn × E k −n ⊂ G k −n E for k n. (See Section 4 for details.)
Proposition 1·3. Let G be either Σ2n or A2n and E % Z/2, then the homomorphism
Res : F2 ⊗ H ∗ (G
A

induced by the inclusion V ⊂ G
k

k −n

k −n

E) → F2 ⊗ H ∗ (Vk )
A

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, and Vk 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 ) %
Pk
F2 [x1 , . . . , xk ] with deg (xi ) = 1. However, they could be read off from Agenerators of the invariant algebra PkW , for some subgroups W of GL(k, F2 ). This is a


Cohomology of the symmetric and the alternating groups

459

motivation for the study of A-generators of PkW , 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 Σm and O1 , O2 , ..., Ot all
the orbits of A. Then we have an inclusion
A ⊂ A|O 1 × A|O 2 × · · · × 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|O 1 × A|O 2 × · · · × 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 an F2 -vector space structure V on O such that C is
isomorphic to (the group of all translations on) the additive group V.
Proof. To make the paper self-contained, we give a proof for this lemma.
Let 0 be a fixed point of O. For any g ∈ C, we set vg = g(0). As C acts faithfully
and transitively on O, we get O = {vg | g ∈ C}. Suppose C is an additive group.
Obviously, O is equipped with an F2 -vector space structure, denoted by V, by setting
vg + vh = vg +h ,
avg = vag ,
for g, h ∈ C, a ∈ F2 .
For every v ∈ V, there is h ∈ C such that v = h(0) = vh . Then, for any g ∈ C, we
have
g(v) = g(h(0)) = (g + h)(0) = vg +h = vg + vh = v + vg .
Thus, g is the translation by vg on V. So, C is a subgroup of (the group of all translations on) the additive group V. Further, since the abelian group C acts faithfully
and transitively on V, we get |C| = |V|. In conclusion, C % V.
As Oi is an orbit of A, it is easily seen that A|O i acts faithfully and transitively on

the orbit Oi for 1 i t. Therefore, by the lemma, we have
A|O i % Vk i
for 1 i t, where Vk i is a vector space of certain dimension ki over F2 .
` [10], the group
According to Mui
A = A|O 1 × A|O 2 × · · · × A|O t % Vk 1 × · · · × Vk t
is a maximal elementary abelian 2-subgroup of Σm = ΣX if and only if there is at
most one of the orbits O1 , . . . , Ot with cardinality 1, or equivalently there is at most
one of the dimensions ki equaling 0.


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Nguye˜ˆ 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-dimensional F2 -vector space V = Vk with |V| = |O| = |X| = 2k . So, the symmetric
group ΣX is isomorphic to Σ2k .
Proposition 2·2. Let V = Vk be an elementary abelian 2-group of rank k > 2. Then,
the homomorphism
Res : F2 ⊗ H ∗ (Σ2k ) −→ F2 ⊗ H ∗ (V)
A

A

induced by the regular permutation representation V ⊂ Σ2k is trivial in positive degrees.
Proof. It is well known that the Weyl group of V in Σ2k is the general linear group
` [10], the image of the restriction
GL(V) = GL(k, F2 ). Further, according to Mui

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 ⊗ H ∗ (Σ2k ) → F2 ⊗ H ∗ (V) factors through
A

A

F2 ⊗ (H ∗ (V)G L (V) ). More precisely, we have a commutative diagram
A

F2 ⊗ H ∗ (Σ2k )
A

R es

✲ F2 ⊗ H ∗ (V)

❅

❅
❅
❘
❅

 
✒
 

A

 
 

F2 ⊗ (H ∗ (V)G L (V) ) ,
A

where the homomorphism F2 ⊗ (H ∗ (V)G L (V) ) → F2 ⊗ H ∗ (V) is induced by the inclusion
A
A
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 ⊗ H ∗ (Σ2k ) → F2 ⊗ H ∗ (V) is zero in positive
A
A
degrees for k > 2.
Remark 2·3. That the map F2 ⊗ (H ∗ (V)G L (V) ) → F2 ⊗ H ∗ (V) is zero in positive deA

A

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 Steenrod 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 Theorem 1·1. From the hypothesis, O is one of the orbits O1 , . . . , Ot and
A|O is one of the summands in the decomposition
A = A|O 1 × A|O 2 × · · · × 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| = 2k , then by Lemma 2·1 A|O 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 ⊂ Σm .
The inclusions show that the homomorphism
Res : F2 ⊗ H ∗ (Σm ) → F2 ⊗ H ∗ (A|O )
A

A

factors through F2 ⊗ H ∗ (Σ2k ). That is, we have a commutative diagram
A

F2 ⊗ H ∗ (Σm )
A

✲ F2 ⊗ H ∗ (A|O )

R es

❅

 
✒

 

❅

A

 

❅
❘
❅

 

F2 ⊗ H ∗ (Σ2k ).
A

By Proposition 2·2, the map
Res : F2 ⊗ H ∗ (Σ2k ) → F2 ⊗ H ∗ (A|O )
A

A

is trivial in positive degrees for k > 2, or equivalently for |O| = 2k > 4. The theorem
follows.
3. The case of the alternating groups
` in [10] and [11] on elementary
Lemmas 3·1 and 3·2 deal with some results by H. Mui
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 that Vk is an elementary abelian 2-group and Vk ⊂ 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
subgroup A2k also acts on Vk .
Lemma 3·1. Vk ⊂ A2k for k

2.

Proof. Each element g ∈ Vk , regarded as a translation on Vk , is of order 2. Thus,
g = (x, g(x)) is a product of 2k −1 transpositions. If k 2, then 2k −1 is even. So, g
is an even permutation. The lemma follows.
The general linear group GLk = GL(Vk ) acts regularly on Vk .
Lemma 3·2. GLk ⊂ A2k for k > 2.
Proof. Choose a basis (e1 , e2 , . . . , ek ) of Vk . Then, GLk can be identified with the
group of all invertible k × k-matrices with entries in F2 . Let Bij be the matrix, whose
entries are zero except the ones on the main diagonal and the one appearing in the
ith row and the jth column (for 1 i, j k, i  j). The group GLk is generated by


462

Nguye˜ˆ n H. V. Hung

{Bij | 1 i  j k}. Indeed, the multiplication of Bij from the left to a matrix B is
the addition of the jth row to the ith row of B; while the multiplication of Bij from
the right to a matrix B is the addition of the ith column to the jth column of B. Each
matrix B ∈ GLk can be transformed into the identity matrix by multiplications with

some matrices Bij either from the left or from the right.
Since i  j, we have Bij (x) = x if and only if x belongs to the (k − 1)-dimensional
vector space Span (e1 , . . . , eˆj , . . . , ek ). As Bij is of order 2, it is a product of 2k −1 /2 =
2k −2 transpositions. Therefore, if k > 2, then Bij is an even permutation. Hence,
GLk ⊂ A2k for k > 2.
Suppose H is a subgroup of G. Let NG (H), CG (H), WG (H) denote respectively the
normalizer, the centralizer and the Weyl group of H in G.
Lemma 3·3. WA2k (Vk ) = GLk for k > 2.
Proof. As is well known, NΣ2k (Vk ) = Vk ×GLk and CΣ2k (Vk ) = Vk (see e.g. [10]).
Using Lemmas 3·1 and 3·2, we have
NA2k (Vk ) = NΣ2k (Vk )  A2k = (Vk ×GLk )  A2k = Vk ×GLk ,
CA2k (Vk ) = CΣ2k (Vk )  A2k = Vk  A2k = Vk
for k > 2. So, we get
WA2k (Vk ) = NA2k (Vk )/CA2k (Vk ) = Vk ×GLk /Vk = GLk
for k > 2.
Lemma 3·4. Let B be a maximal elementary abelian 2-subgroup of An . If O is an orbit
4, then B|O ⊂ A|O | . Further, there exists an F2 -vector
of B with cardinality |O| = 2k
space structure Vk on O such that B|O is isomorphic to (the group of all translations on)
the additive group of Vk .
Proof. As O is an orbit of B, it is easy to see that B|O acts faithfully and transitively
on O. Then, by Lemma 2·1, there exists an F2 -vector space structure Vk on O such
that B|O is isomorphic to (the group of all translations on) the additive group of Vk .
On the other hand, by Lemma 3·1, Vk ⊂ A2k = A|O | for k 2.
Proposition 3·5. Let V = Vk be an elementary abelian 2-group of rank k > 2. Then,
the homomorphism
Res : F2 ⊗ H ∗ (A2k ) → F2 ⊗ H ∗ (V)
A

A

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 ) =
GLk 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 ⊗ H ∗ (A2k ) → F2 ⊗ H ∗ (V)
A

∗

factors through F2 ⊗ (H (V)
homomorphism

G L (V)

A

A

). The proposition follows from the fact that the

F2 ⊗ H ∗ (V)G L (V) → F2 ⊗ H ∗ (V)
A

A


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.
Proof of Theorem 1·2. The proof is similar to that of Theorem 1·1. As B is a
2-group, the cardinality of O is a power of 2. From the hypothesis, |O| = 2k > 4,
thus k > 2. Then, by Lemmas 3·1 and 3·4, we get
B|O % V = Vk ⊂ A2k ⊂ Am .
The homomorphism
Res : F2 ⊗ H ∗ (Am ) → F2 ⊗ H ∗ (B|O ) % F2 ⊗ H ∗ (V)
A

A

A

factors through Res : F2 ⊗ H ∗ (A2k ) → F2 ⊗ H ∗ (V). By Proposition 3·5, the last
A

A

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

1

E = G E, G

k

E= G

k −1

E

E.

One regards G × E as a subgroup of G E via the inclusion
G × E ⊂ G E = (G × G)×E,
(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 GLn and K is a subgroup of GLk −n for k n. We are
interested in the following group
H •K

A ∗
0 B

A ∈ H, B ∈ K

GLk ,

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 1k −n or the Sylow
2-subgroup Tk −n of GLk −n consisting of all upper triangular matrices with 1 on the

main diagonal.
Lemma 4·1. Let Vn G Σ2n , where the inclusion Vn ⊂ Σ2n is the regular permutation representation of Vn . Then the Weyl group WG (Vn ) is isomorphic to a subgroup of
GLn % GL(Vn ) and
WG E (V n × E) % WG (Vn ) • 11

GLn +1 .

Proof. By definition, σ ∈ CG (V ) if and only if σtσ −1 = t for every t ∈ Vn . In
particular, we have
n

σt(0) = tσ(0),
σ(t(0)) = σ(0) + t(0).


464

Nguye˜ˆ n H. V. Hung

As V acts transitively on itself by translation, each element u ∈ Vn can be written
in the form u = t(0), for some t ∈ Vn . Hence, σ is nothing but the translation by σ(0)
on Vn . Therefore, CG (Vn ) ⊂ Vn . On the other hand, as Vn is abelian, Vn ⊂ CG (Vn ).
In summary, we get CG (Vn ) = Vn . By similarity, CG E (Vn × E) = Vn × E.
Let us consider the conjugacy homomorphism
n

j = jG : NG (Vn ) → Aut(Vn ) % GLn ,
g → (cg : v → gvg −1 ).
By definition of the centralizer subgroup, Ker(jG ) = CG (Vn ) = Vn . Thus
WG (Vn ) = NG (Vn )/CG (Vn ) = NG (Vn )/Ker(jG ) % Im(jG ).

Therefore, WG (Vn ) is isomorphic to a subgroup of GLn . Similarly, we get
WG E (Vn × E) = NG E (Vn × E)/Ker(jG E ) % Im(jG E ).
So, in order to determine WG E (Vn × E), we need to compute the image of jG E .
Let us divide the argument into two steps.
Step 1: we find the conditions for σ = (g, h; 0) ∈ NG E (Vn × E), where g, h ∈ G.
An element of V n × E, regarded as a subgroup of G E, is either (v, v; 0) or (v, v; 1),
where v ∈ Vn 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 to Vn × E if and only if
gvg −1 ∈ Vn , hvh−1 ∈ Vn , gvg −1 = hvh−1 ,
for every v ∈ Vn . The first two conditions are respectively equivalent to g ∈ NG (Vn )
and h ∈ NG (Vn ). The last condition is equivalent to
v = g −1 hvh−1 g = (g −1 h)v(g −1 h)−1 ,
for every v ∈ Vn . That is g −1 h ∈ Vn = CG (Vn ).
Notice that if g ∈ NG (Vn ) and g −1 h ∈ Vn , then g(g −1 h)g −1 = hg −1 ∈ Vn . Hence
(hg −1 )−1 = gh−1 ∈ Vn .
We now show that under the hypotheses g, h ∈ NG (Vn ) and g −1 h ∈ Vn , we get
σ(v, v; 1)σ −1 = (g, h; 0)(v, v; 1)(g, h; 0)−1 = (gvh−1 , hvg −1 ; 1) ∈ Vn × E,
for every v ∈ Vn . Indeed, this is equivalent to
gvh−1 ∈ Vn , hvg −1 ∈ Vn , gvh−1 = hvg −1 ,
for every v ∈ Vn . The first condition satisfies because of
gvh−1 = (gvg −1 )(gh−1 )
and of (gvg −1 ) ∈ Vn , gh−1 ∈ Vn . Similarly, the second condition also satisfies. Since
g −1 h, v ∈ Vn and Vn is an elementary abelian 2-group, we have
g −1 hvg −1 h = (g −1 h)2 v = v.
This is equivalent to the third condition.
In summary, we have shown that σ = (g, h; 0) ∈ NG E (Vn × E) if and only if
g, h ∈ NG (Vn ) and g −1 h ∈ Vn .

Cohomology of the symmetric and the alternating groups

465

Note that V is an additive group, while G is a multiplicative one. More precisely,
V is regarded as a subgroup of G via a group monomorphism
n

n

⊂

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 → G E. If g ∈ NG (Vn ), then by jG (g) = A ∈ GLn %
Aut(Vn ) we mean
i−1 (gvg −1 ) = Ai−1 (v),
for every v ∈ Vn .
For σ = (g, h; 0) ∈ NG E (Vn × E), we set A = jG (g) ∈ GLn and b = i−1 (gh−1 ) ∈ Vn .
As gvg −1 , gh−1 ∈ Vn , 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) ∈ NG 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
jG E (σ) = jG E (g, h; 0) =

A b

0 1

∈ GLn +1 .

Step 2: similarly, σ = (g, h; 1) ∈ NG E (Vn × E) if and only if g, h ∈ NG (Vn ) and
g −1 h ∈ Vn .
Then, setting A = jG (g) ∈ GLn and b = i−1 (gh−1 ) ∈ Vn , 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,
jG E (σ) = jG E (g, h; 1) =

A b
0 1

∈ GLn +1 .

Conversely, given A ∈ Im(jG ) and b ∈ Vn , there is g ∈ NG (Vn ) such that jG (g) = A.
Define h by the equality b = i−1 (gh−1 ), we easily show that h ∈ NG (Vn ) and that
jG E (g, h; 0) = jG E (g, h; 1) =

A b
.
0 1

In conclusion, we have shown that
Im(jG E ) =

A b
0 1

A ∈ Im(jG ) % WG (Vn ), b ∈ Vn .

In other words,
WG E (Vn × E) % Im(jG E ) % WG (Vn ) • 11 .
The lemma is proved.


466

Nguye˜ˆ n H. V. Hung

Suppose Vn is a subgroup of G. Then Vk % Vn × E k −n is a subgroup of G × E k −n ,
therefore it is a subgroup of G k −n E for k n.
Corollary 4·2. Let G be either Σ2n or A2n and Vn ⊂ G the regular permutation
representation of Vn . Then the Weyl group of Vk in G k −n E is given by
WG k −n E (Vk ) % GLn • Tk −n ,
for k n > 2. Here Tk −n denotes the Sylow 2-subgroup of GLk −n consisting of all upper
triangular matrices with 1 on the main diagonal.
Proof. It is shown by induction on k. From Lemma 3·1, the image of the regular
permutation representation Vn ⊂ Σ2n is a subgroup of A2n for n 2. We know that
WΣ2n (Vn ) % WA 2n (Vn ) % GLn
for n > 2. (See e.g. [10] and Lemma 3·3.) So, the corollary holds for k = n > 2.
Suppose inductively that
WG k −n E (Vk ) % GLn • Tk −n .
Then, by Lemma 4·1, we have
WG k +1−n E (Vk +1 ) % WG k +1−n E (Vk × E) % WG k −n E (Vk ) • 11
% (GLn • Tk −n ) • 11 = GLn • Tk +1−n .
The purpose of this section is to show the following, which is also numbered as
Proposition 1·3 in the introduction.

Proposition 4·3. Let G be either Σ2n or A2n and Vn ⊂ G the regular permutation
representation of Vn . Then the homomorphism
Res : F2 ⊗ H ∗ (G

k −n

A

induced by the inclusion V ⊂ G
k

k −n

E) → F2 ⊗ H ∗ (Vk )
A

E is trivial in positive degrees for k

n > 2.

Proof. The restriction homomorphism H ∗ (G k −n E) → H ∗ (Vk ) factors through
the invariant algebra H ∗ (Vk )W , where, by Corollary 4·2, W % GLn • Tk −n is the
Weyl group of Vk in the iterated wreath product G k −n E. Therefore, Res factors
through the homomorphism
F2 ⊗ (H ∗ (Vk )G L n •T k −n ) → F2 ⊗ H ∗ (Vk )
A

A

∗

k G L n •T k −n

∗

⊂ H (V ). As GLn • 1k −n is a subgroup of
induced by the inclusion H (V )
GLn • Tk −n , the above homomorphism again factors through the one
k

F2 ⊗ (H ∗ (Vk )G L n •1k −n ) → F2 ⊗ H ∗ (Vk )
A
∗

A

k G L n •1k −n

∗

⊂ H (V ). The last homomorphism is shown
induced by the inclusion H (V )
.
by Hu ng–Nam [7] to be zero in positive degrees for k n > 2.
k

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