p-Groups with a unique proper non-trivial characteristic subgroup

Glasby, S. P.; Pálfy, P. P.; Schneider, Csaba

doi:10.1016/j.jalgebra.2011.10.005

Cited by 11 publications

(15 citation statements)

References 16 publications

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“…For previous usages of this technique, see [HL74,DH75,Car83,GPS11]. Namely, both V = G/G ′ and G ′ can be regarded as vector spaces over the field GF(p) with p elements.…”

Section: Introductionmentioning

confidence: 99%

A module-theoretic approach to abelian automorphism groups

Caranti

2014

Isr. J. Math.

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Abstract. There are several examples in the literature of finite non-abelian p-groups whose automorphism group is abelian. For some time only examples that were special p-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav [JRY13] constructed several non-special examples. In this paper we show how a simple module-theoretic approach allows the construction of non-special examples, starting from special ones constructed by several authors, while at the same time avoiding further direct calculations.

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“…For previous usages of this technique, see [HL74,DH75,Car83,GPS11]. Namely, both V = G/G ′ and G ′ can be regarded as vector spaces over the field GF(p) with p elements.…”

Section: Introductionmentioning

confidence: 99%

A module-theoretic approach to abelian automorphism groups

Caranti

2014

Isr. J. Math.

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“…A group G with a unique non-trivial proper characteristic subgroup is abbreviated a UCS group. Finite UCS groups were studied by Taunt [Tau55] and later finite UCS p-groups were explored by the first and the third authors in collaboration with P. P. Pálfy [GPS11]. The characteristic subgroups of a finite UCS p-group G are 1, Φ(G), and G, and consequently the exponent of G is either p or p 2 .…”

Section: Ucs P-groups Of Exponent Pmentioning

confidence: 99%

“…Lemma 2.1. [GPS11,Lemma 3]. Suppose that G is a finite non-abelian UCS p-group and 1 ✁ N ✁ G are the only characteristic subgroups of G. Then the following hold:…”

Section: Ucs P-groups Of Exponent Pmentioning

confidence: 99%

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Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras

Glasby¹,

Ribeiro

Schneider

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let p be an odd prime and let G be a non-abelian finite p-group of exponent p 2 with three distinct characteristic subgroups, namely 1, G p , and G. The quotient group G/G p gives rise to an anti-commutative F p -algebra L such that the action of Aut(L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, F).

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“…products of isomorphic simple groups have no proper nontrivial characteristic subgroups. Taunt and Glasby-Pálfy-Schneider characterized groups with a unique proper nontrivial characteristic subgroup [12,29]. Those situations seem rare when compared to the complexity of general finite groups.…”

Section: Introductionmentioning

confidence: 99%

More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

Wilson¹

2013

Journal of Group Theory

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Abstract. We introduce three families of characteristic subgroups that refine the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. We prove that a positive logarithmic proportion of finite p-groups admit at least five such proper nontrivial characteristic subgroups whereas verbal and marginal methods explain only one. The placement of these subgroups in the lattice of subgroups is naturally recorded by a filter over an arbitrary commutative monoid M and induces an M -graded Lie ring. These Lie rings permit an efficient specialization of the nilpotent quotient algorithm to construct automorphisms and decide isomorphism of finite p-groups. As a demonstration, we identify some large families of p-groups that are worst-case examples for the traditional nilpotent quotient algorithm but run in polynomial time when using our new filters.

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p-Groups with a unique proper non-trivial characteristic subgroup

Cited by 11 publications

References 16 publications

A module-theoretic approach to abelian automorphism groups

A module-theoretic approach to abelian automorphism groups

Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras

More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

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