Finite self dual groups

An, Lijian; Ding, Jianfang; Zhang, Qinhai

doi:10.1016/j.jalgebra.2011.06.014

Cited by 7 publications

(7 citation statements)

References 4 publications

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“…If y = 0, then the equation x 2 + ry 2 − 1 = 0 is equivalent to (xy −1 ) 2 − y −2 + r = 0. By (1), the later has p − 1 solutions. Hence n(1) = p + 1.…”

Section: Preliminariesmentioning

confidence: 99%

Finite $p$-groups all of whose subgroups of index $p^3$ are abelian

Zhang

Zhao

et al. 2014

Preprint

Self Cite

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Suppose that G is a finite p-group. If all subgroups of index p t of G are abelian and at least one subgroup of index p t−1 of G is not abelian, then G is called an A t -group. In this paper, some information about A t -groups are obtained and A 3groups are completely classified. This solves an old problem proposed by Berkovich and Janko in their book [8]. Abundant information about A 3 -groups are given.Keywords finite p-groups, minimal non-abelian p-groups, A t -groups. 2010MSC 20D15

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“…If y = 0, then the equation x 2 + ry 2 − 1 = 0 is equivalent to (xy −1 ) 2 − y −2 + r = 0. By (1), the later has p − 1 solutions. Hence n(1) = p + 1.…”

Section: Preliminariesmentioning

confidence: 99%

Finite $p$-groups all of whose subgroups of index $p^3$ are abelian

Zhang

Zhao

et al. 2014

Preprint

Self Cite

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“…Such a finite group is called a s-self dual group and these groups have been completely classified in [1]. This classification involves two families of finite p-groups.…”

Section: Some Non-vanishing Groupsmentioning

confidence: 99%

“…Let p be a prime number, then we denote by X p 3 an extra-special p-group of exponent p. Let n be an integer, then we denote by M p (n, n) the finite group <a, b | a p n = b p n = 1, bab −1 = a 1+p n−1 >. The following theorem summarizes the classification in [1].…”

Section: Some Non-vanishing Groupsmentioning

confidence: 99%

“…So his method cannot be generalized to the case where the characteristic of the field is p. Let us look more carefully at the action of kB(G, G) on kB(G). Let H be a subgroup of G × G and L be a subgroup of G. Then, the action of the transitive G-G biset (G × G)/H on the transitive G-set G/L is given by the Mackey formula (see formula (1)). Since G is a commutative group, the action is given by:…”

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confidence: 99%

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Around evaluations of biset functors

Rognerud¹

2019

Annales De l'Institut Fourier

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The evaluation functor carries important information about the category of biset functors into the category of modules over the double Burnside algebra. Our purpose here, is to study double Burnside algebras via evaluations of biset functors. In order to avoid the difficult problem of vanishing of simple functors, we look at finite groups for which there is no non-trivial vanishing and we call them non-vanishing groups. This family contains all the abelian groups, but also infinitely many others. We show that for a non-vanishing group, there is an equivalence between the category of modules over the double Burnside algebra and a specific category of biset functors. Then, we deduce results about the highest-weight structure, and the self-injective property of the double Burnside algebra. We also revisit Barker's Theorem on the semi-simplicity of the category of biset functors.

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“…Spencer [8], and studied more recently by L. An, J. Ding and Q. Zhang [9]. A group is said to be self dual if the isomorphism classes of its subgroups and of its quotient groups coincide.…”

Section: Every Homomorphic Image Of G Is Isomorphic To a Normal Subgrmentioning

confidence: 99%

Finite morphic p-groups

Caranti

Scoppola²

2015

Journal of Pure and Applied Algebra

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According to Li, Nicholson and Zan (2010), a group G is said to be morphic if, for every pair N 1 , N 2 of normal subgroups, each of the conditions G/N 1 ∼ = N 2 and G/N 2 ∼ = N 1 implies the other. Finite, homocyclic p-groups are morphic, and so is the nonabelian group of order p 3 and exponent p, for p an odd prime. It follows from results of An, Ding and Zhan (2011) on self dual groups that these are the only examples of finite, morphic p-groups. In this paper we obtain the same result under a weaker hypothesis.

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Finite self dual groups

Cited by 7 publications

References 4 publications

Finite $p$-groups all of whose subgroups of index $p^3$ are abelian

Finite $p$-groups all of whose subgroups of index $p^3$ are abelian

Around evaluations of biset functors

Finite morphic p-groups

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