Finite groups with seminormal Sylow subgroups

Guo, Wen Bin

doi:10.1007/s10114-008-6563-z

Cited by 8 publications

(5 citation statements)

References 3 publications

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“…Groups with some seminormal subgroups have been investigated in the works of many authors, see, for example, [6,14,16,21,22,25]. We need the following properties.…”

Section: Groups With Some Seminormal Subgroups In Maximal Subgroupsmentioning

confidence: 99%

Finite groups with two supersoluble subgroups

Монахов

Trofimuk

2018

Journal of Group Theory

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LetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

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“…Groups with some seminormal subgroups have been investigated in the works of many authors, see, for example, [6,14,16,21,22,25]. We need the following properties.…”

Section: Groups With Some Seminormal Subgroups In Maximal Subgroupsmentioning

confidence: 99%

Finite groups with two supersoluble subgroups

Монахов

Trofimuk

2018

Journal of Group Theory

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“…Corollary 5.4 (See [36,Theorem]). Let P be a Sylow p-subgroup of G. If P is semi-normal in G, then the following statements hold:…”

Section: Groups With a Hall Quasipermutable Subgroupmentioning

confidence: 99%

Some new characterizations of $PST$-groups

Skiba

2013

Preprint

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Let H and B be subgroups of a finite group G such that G = N G (H)B. Then we say that H is quasipermutable (respectively S-quasipermutable) in G provided H permutes with B and with every subgroup (respectively with every Sylow subgroup) A of B such that (|H|, |A|) = 1. In this paper we analyze the influence of S-quasipermutable and quasipermutable subgroups on the structure of G. As an application, we give new characterizations of soluble P ST -groups. IntroductionThroughout this paper, all groups are finite and G always denotes a finite group. Moreover p is always supposed to be a prime and π is a subset of the set P of all primes; π(G) denotes the set of all primes dividing |G|.A subgroup H of G is said to be quasinormal or permutable in G if H permutes with every subgroup A of G, that is, HA = AH; H is said to be S-permutable in G if H permutes with every Sylow subgroup of G.As well as T -groups, P T -groups and P ST -groups possess many interesting properties (see Chapter 2 in [1]). The general description of P T -groups and P ST -groups were firstly obtained by Zacher [2] and Agrawal [3], for the soluble case, and by Robinson in [4], for the general case. Nevertheless,

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“…Groups with some seminormal subgroups have been investigated by many authors; see, for example, [7,11,15,17,23]. In particular, the supersolubility of a group with seminormal Sylow subgroups was obtained in [11,17].…”

Section: Introductionmentioning

confidence: 99%

On the supersolubility of a group with semisubnormal factors

Монахов

Trofimuk

2020

Journal of Group Theory

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A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that {G=AB} and AX is a subgroup of G for every subgroup X of B. We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semisubnormal in G if A is subnormal in G or seminormal in G. A group {G=AB} with semisubnormal supersoluble subgroups A and B is studied. The equality {G^{\mathfrak{U}}=(G^{\prime})^{\mathfrak{N}}} is established; moreover, if the indices of subgroups A and B in G are relatively prime, then {G^{\mathfrak{U}}=G^{\mathfrak{N}^{2}}}. Here {\mathfrak{N}}, {\mathfrak{U}} and {\mathfrak{N}^{2}} are the formations of all nilpotent, supersoluble and metanilpotent groups, respectively; {H^{\mathfrak{X}}} is the {\mathfrak{X}}-residual of H. Also we prove the supersolubility of {G=AB} when all Sylow subgroups of A and of B are semisubnormal in G.

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Finite groups with seminormal Sylow subgroups

Cited by 8 publications

References 3 publications

Finite groups with two supersoluble subgroups

Finite groups with two supersoluble subgroups

Some new characterizations of $PST$-groups

On the supersolubility of a group with semisubnormal factors

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