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

DSpace at VNU: On A-generators for the cohomology of the symmetric and the alternating groups tài liệu, giáo án, bài giả...

Mathematical Proceedings of the Cambridge Philosophical Society http://journals.cambridge.org/PSP Additional services for Mathematical Proceedings of the Cambridge Philosophical Society: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here 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 Request Permissions : Click here Downloaded from http://journals.cambridge.org/PSP, IP address: 132.239.1.230 on 15 Mar 2015 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: nhvhung@vnu.edu.vn (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 Introduction and statement of results Throughout this paper, by the cohomology H ∗ (G) of a finite group G we mean the mod 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 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 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| > 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| > 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 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 > 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| = or and so does the proposition for n 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 ) = 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 and Section Finally, the case of the wreath products is investigated in Section 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 × A|O × · · · × 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 × A|O × · · · × 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 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 i t Therefore, by the lemma, we have A|O i % Vk i for i t, where Vk i is a vector space of certain dimension ki over F2 ` [10], the group According to Mui A = A|O × A|O × · · · × A|O t % Vk × · · · × 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 460 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 > 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 > As a consequence, the map Res : F2 ⊗ H ∗ (Σ2k ) → F2 ⊗ H ∗ (V) is zero in positive A A degrees for k > Remark 2·3 That the map F2 ⊗ (H ∗ (V)G L (V) ) → F2 ⊗ H ∗ (V) is zero in positive deA A grees for k > is equivalent to the fact that the Dickson algebra H ∗ (V)G L (V) with dim (V) > 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) = or 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 × A|O × · · · × 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 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 > The theorem follows 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 Proof Each element g ∈ Vk , regarded as a translation on Vk , is of order 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 > 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 i, j k, i  j) The group GLk is generated by 462 Nguye˜ˆ n H V Hung {Bij | 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 > 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 > 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 > So, we get WA2k (Vk ) = NA2k (Vk )/CA2k (Vk ) = Vk ×GLk /Vk = GLk for k > 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 Proposition 3·5 Let V = Vk be an elementary abelian 2-group of rank k > 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 > 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 > (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 From the hypothesis, |O| = 2k > 4, thus k > 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 > 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 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 ∗ 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 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, ∈ 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 ∈ 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 ∈ 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 In conclusion, we have shown that Im(jG E ) = A b 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 > Here Tk −n denotes the Sylow 2-subgroup of GLk −n consisting of all upper triangular matrices with 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 We know that WΣ2n (Vn ) % WA 2n (Vn ) % GLn for n > (See e.g [10] and Lemma 3·3.) So, the corollary holds for k = n > 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 > 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 > k REFERENCES [1] E B Curtis The Dyer–Lashof algebra and the Λ-algebra Illinois J Math 18 (1975), 231–246 [2] Nguye˜ˆ n H V Hu.ng Spherical classes and the algebraic transfer Trans Amer Math Soc 349 (1997), 3893–3910 Cohomology of the symmetric and the alternating groups [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] 467 Nguye˜ˆ n H V Hu.ng The weak conjecture on spherical classes Math Zeit 231 (1999), 727– 743 Nguye˜ˆ n H V Hu.ng, Spherical classes and the lambda algebra Trans Amer Math Soc 353 (2001), 4447–4460 Nguye˜ˆ n H V Hu.ng On triviality of Dickson invariants in the homology of the Steenrod algeb Math Proc Camb Phil Soc 134 (2003), 103–113 Nguye˜ˆ n H V Hu.ng and Tr`aˆ n N Nam The hit problem for the Dickson algebra Trans Amer Math Soc 353 (2001), 5029–5040 Nguye˜ˆ n H V Hu.ng and Tr`aˆ n N Nam The hit problem for the modular invariants of linear groups J Algebra 246 (2001), 367–384 Nguye˜ˆ n H V Hu.ng and F P Peterson A–generators for the Dickson algebra Trans Amer Math Soc 347 (1995), 4687–4728 Nguye˜ˆ n H V Hu.ng and F P Peterson Spherical classes and the Dickson algebra Math Proc Camb Phil Soc 124 (1998), 253–264 Huynh ` Mui ` Modular invariant theory and cohomology algebras of symmetric groups J Fac Sci Univ Tokyo 22 (1975), 310–369 Huynh ` Mui ` Cohomology operations derived from modular invariants Math Zeit 193 (1986), 151–163 ` M Quang Classification of maximal elementary abelian 2-subgroups of the alternating Hoang groups BS Thesis, Vietnam National University (2003), 29 pages (in Vietnamese) D Quillen The spectrum of an equivariant cohomology ring I, II Ann of Math 94 (1971), 449–602 V Snaith and J Tornehave On π∗S (BO) and the Arf invariant of framed manifolds Amer Math Soc Contemp Math 12 (1982), 299–313  ... the symmetric groups and that of the alternating groups are respectively studied in Sections and Section Finally, the case of the wreath products is investigated in Section The case of the symmetric. .. 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. .. 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