The integral cohomology rings of some <i>p</i>-groups

Leary, Ian J.

doi:10.1017/s0305004100070080

Cited by 32 publications

(48 citation statements)

References 12 publications

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“…Then in section 3, we shall use Massey products to define some generators of degree two in the mod-p cohomology ring of G and to determine some of the relations involving these generators. This explicit use of Massey products to obtain the cohomology ring structure has been demonstrated by Leary in [3] and [5]. We remark here that the mod-p cohomology ring structure of G…”

supporting

confidence: 56%

“…Let p denote an odd prime and let P m n be the group with presentation of the form where C = <C> = C p and L = (A, B} = C p m X C p n. In this paper we shall determine the mod-p cohomology ring and some of the integral cohomology ring structure of P m n when m,n>l. For the case when m = n = 1, P ltl is the non-abelian group of order p 3 and exponentp, and the integral and mod-p cohomology rings of P l { are known (see [4], [5], [7]). We note that for £ = 3 and m, n > 1, Leary in [6] has obtained the Poincare series of H*(P m n , F 3 ).…”

Section: § 1 Introductionmentioning

confidence: 99%

“…In [3,5] , Leary showed how Massey products can be used to define some generators of low degrees in a mod-p cohomology ring and to obtain some of the relations involving these generators. Leary went on to show in [6] that the image of the differential d± in the Lyndon-Hochschild-Serre (LHS) spectral sequence (with W p coefficients) of certain central extensions involves a Massey product.…”

Section: § 1 Introductionmentioning

confidence: 99%

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The Cohomology Rings of Some $p$-Groups

Chin

1995

Publ. Res. Inst. Math. Sci.

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We determine the mod-p cohomology ring and some of the integral cohomology ring structure of a p-group expressible as an extension with kernel cyclic of order p and quotient C p m 0 C p n, where m, n > 1. § 1. IntroductionLet p denote an odd prime and let P m n be the group with presentation of the formwhere m, n > 1. We may express P m n as a central extension of the formwhere C = = C p and L = (A, B} = C p m X C p n. In this paper we shall determine the mod-p cohomology ring and some of the integral cohomology ring structure of P m n when m,n>l. For the case when m = n = 1, P ltl is the non-abelian group of order p 3 and exponentp, and the integral and mod-p cohomology rings of P l { are known (see [4], [5], [7]). We note that for £ = 3 and m, n > 1, Leary in [6] has obtained the Poincare series of H*(P m n , F 3 ). Let G = P m> " where m, n > 1. It is clear that we may assume, without loss of generality, that m > n > 2. In section 2 of this paper we shall review some facts on Massey products. Then in section 3, we shall use Massey products to define some generators of degree two in the mod-p cohomology ring of G and to determine some of the relations involving these generators. This explicit use of Massey products to obtain the cohomology ring structure has been demonstrated by Leary in [3] and [5]. We remark here that the mod-p cohomology ring structure of G

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confidence: 56%

Section: § 1 Introductionmentioning

confidence: 99%

Section: § 1 Introductionmentioning

confidence: 99%

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The Cohomology Rings of Some $p$-Groups

Chin

1995

Publ. Res. Inst. Math. Sci.

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“…Conversely a direct proof of this fact would yield a new proof of Minh's result. If n = 1 and p > 3 it is known (see [8]) that c 2 (ρ) is a nonzero essential class which annihilates κ 1 .…”

Section: An Examplementioning

confidence: 99%

On Carlson's depth conjecture in group cohomology

Green

2003

Mathematische Zeitschrift

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We establish a weak form of Carlson's conjecture on the depth of the mod-p cohomology ring of a p-group. In particular, Duflot's lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure.Comment: 10 page

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“…We can take products of periodic groups G 1 × G 2 and obtain a variety of actions of non-metacyclic groups on S n × S n (see [16] for the existence of these examples, generalizing the results of Stein [34]). One of the key points is that the cohomology of the group Γ is much simpler than that of its finite subgroups (see Leary [24] for Γ, and Lewis [28] for the extra-special p-groups), so the computations of Steps (i) and (ii) are best done over Γ. However, there are non-metabelian 2-groups which are subgroups of 3292 IAN HAMBLETON ANDÖZGÜNÜNLÜ (i) We construct a non-empty subset T Γ ⊆ H 2n−1 (B Γ ; Z), depending on the data (θ 1 , θ 2 , ν Γ ), consisting entirely of primitive elements of infinite order.…”

Section: Introductionmentioning

confidence: 99%

Free actions of finite groups on $S^n \times S^n$

Hambleton

Ünlü

2009

Trans. Amer. Math. Soc.

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Let p be an odd prime. We construct a non-abelian extension Γ of S 1 by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S 2p−1 × S 2p−1 . In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S 2p−1 × S 2p−1 . These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres. download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use IAN HAMBLETON ANDÖZGÜNÜNLÜThe finite subgroups of Γ which surject onto the quotient Z/p × Z/p are direct products G = C × P (k), where C is a finite cyclic group of order prime to p, andis a rank two p-group of order p k , k ≥ 3. We therefore obtain infinitely many actions of non-metacyclic p-groups on S 2p−1 × S 2p−1 for each prime p. An important special case is the extraspecial p-group G p = P (3) of order p 3 and exponent p. Our existence result contradicts claims made in [4], [5], [37], and [41] that G p -actions do not exist (for cohomological reasons) on any product of equidimensional spheres. It was later shown by Benson and Carlson [7] that such actions could not be ruled out for any prime p by cohomological methods. Moreover for p = 3, in [17], we gave an explicit construction of a free smooth action of Γ (and in particular G 3 ) on S 5 × S 5 . This construction provides an alternate proof of Theorem A for p = 3.More generally, rank two finite p-groups were classified by Blackburn [8] (see also [26]). Consider the additional family, extending the groups P (k):where k ≥ 4, and is 1 or a quadratic non-residue mod p. Here is Blackburn's list of the rank two p-groups G with order p k , and p > 3 (the classification for p = 3 is more complicated):We now know that groups of types I and II do act freely on a product of equidimensional spheres in the minimal dimension. Is this the complete answer?Conjecture. Let p > 3 be an odd prime. If G is a rank two p-group G which acts freely and smoothly on S 2pr−1 × S 2pr−1 , r ≥ 1, then G is metacyclic or G is a subgroup of Γ. Licensed to Univ of Mississippi. Prepared on Sun Jul 5 03:53:55 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use FREE ACTIONS OF FINITE GROUPS ON S n × S n 3291 Sp (2); hence by generalizing the notion of fixity in [2] to quaternionic fixity, one can construct free actions of these non-metabelian 2-groups on S 7 × S 7 (see [38]).Remark. Every rank two finite p-group (for p odd) admits a free smooth action on some product S n × S m , m n (see [2] for p > 3, [38] for p = 3). The survey article by A. Adem [1] describes recent progress on the existence problem in this setting for general finite groups (see also [20]). In most cases, construction of the actions requires m > n.We will always assume that our actions on S n × S n are homologically trivial and n is odd. For free actions...

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The integral cohomology rings of some p-groups

Cited by 32 publications

References 12 publications

The Cohomology Rings of Some $p$-Groups

The Cohomology Rings of Some $p$-Groups

On Carlson's depth conjecture in group cohomology

Free actions of finite groups on $S^n \times S^n$

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