A characterization of powerful p-groups

González-Sánchez, Jon; Zugadi-Reizabal, Amaia

doi:10.1007/s11856-014-1072-5

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1]) a Finite P-group G Is Modular If And Only If2

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2015

2024

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Cited by 2 publications

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“…Let H be a subgroup of G such that d(H) = d(G). Then H is regular and, by a result of Hall [5], |H : H p | = |Ω 1 (H)|. By Theorems 2.3 and 2.7, we have d( …”

Section: 1]) a Finite P-group G Is Modular If And Only Ifmentioning

confidence: 83%

“…(a) G contains an abelian normal subgroup H such that G/H is cyclic and there exist an element g ∈ G with G = g H and a positive integer s such that The following result, conjectured by Klopsch and Snopce in [11], was proved by González-Sánchez and Zugadi-Reizabal in [4]. It is worth noting that for regular finite p-groups the statement of Theorem 1.3 also holds for the prime p = 3, as the following proposition shows.…”

Section: 1]) a Finite P-group G Is Modular If And Only Ifmentioning

confidence: 99%

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Finite p-groups all of whose d-generated subgroups are powerful

Snopce

2015

Journal of Algebra

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“…Let H be a subgroup of G such that d(H) = d(G). Then H is regular and, by a result of Hall [5], |H : H p | = |Ω 1 (H)|. By Theorems 2.3 and 2.7, we have d( …”

Section: 1]) a Finite P-group G Is Modular If And Only Ifmentioning

confidence: 83%

Section: 1]) a Finite P-group G Is Modular If And Only Ifmentioning

confidence: 99%

Finite p-groups all of whose d-generated subgroups are powerful

Snopce

2015

Journal of Algebra

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On groups with large verbal quotients

Lisi,

Sabatini

2024

Journal of Group Theory

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Suppose that w = w ⁢ ( x 1 , … , x n ) w=w(x_{1},\ldots,x_{n}) is a word, i.e. an element of the free group F = ⟨ x 1 , … , x n ⟩ F=\langle x_{1},\ldots,x_{n}\rangle . The verbal subgroup w ⁢ ( G ) w(G) of a group 𝐺 is the subgroup generated by the set { w ⁢ ( x 1 , … , x n ) : x 1 , … , x n ∈ G } \{w(x_{1},\ldots,x_{n}):x_{1},\ldots,x_{n}\in G\} of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if | H : w ( H ) | < | G : w ( G ) | \lvert H:w(H)\rvert<\lvert G:w(G)\rvert for every H < G H<G . In this paper, we give new results on 𝑤-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size 𝑛, then it has a solvable (resp. nilpotent) subgroup of size at least 𝑛.

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A characterization of powerful p-groups

Cited by 2 publications

References 13 publications

Finite p-groups all of whose d-generated subgroups are powerful

Finite p-groups all of whose d-generated subgroups are powerful

On groups with large verbal quotients

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