On the F-hypercentral subgroups with the sylow tower property of finite groups

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

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Finite Groups with Prescribed Sylow Tower Subgroups

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-16.1.577 ◽

1966 ◽

Vol s3-16 (1) ◽

pp. 577-589 ◽

Cited By ~ 4

Author(s):

John S. Rose

Keyword(s):

Finite Groups ◽

Sylow Tower

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A NOTE ON NORMAL COMPLEMENTS FOR FINITE GROUPS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000151 ◽

2018 ◽

Vol 98 (1) ◽

pp. 109-112

Author(s):

NING SU ◽

ADOLFO BALLESTER-BOLINCHES ◽

HANGYANG MENG

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Prime Order ◽

Hall Subgroup ◽

A Finite Group ◽

Sylow Tower

Assume that $G$ is a finite group and $H$ is a 2-nilpotent Sylow tower Hall subgroup of $G$ such that if $x$ and $y$ are $G$-conjugate elements of $H\cap G^{\prime }$ of prime order or order 4, then $x$ and $y$ are $H$-conjugate. We prove that there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N=1$.

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Some generalizations of Shao and Beltrán’s theorem

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500676 ◽

2021 ◽

Author(s):

Minghui Li ◽

Jiakuan Lu ◽

Boru Zhang ◽

Wei Meng

Keyword(s):

Finite Groups ◽

Sylow Subgroup ◽

Invariant Subgroup ◽

Sylow Tower

Let [Formula: see text] and [Formula: see text] be finite groups of relative coprime orders and [Formula: see text] act on [Formula: see text] via automorphisms. In this paper, we prove that when every maximal [Formula: see text]-invariant subgroup of [Formula: see text] that contains the normalizer of some Sylow subgroup has prime index, then [Formula: see text] is supersolvable; if every non-nilpotent maximal [Formula: see text]-invariant subgroup of [Formula: see text] has prime index or is normal in [Formula: see text], then [Formula: see text] is a Sylow tower group.

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GROUPS OF FINITE NORMAL LENGTH

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717001101 ◽

2018 ◽

Vol 97 (2) ◽

pp. 229-239

Author(s):

FRANCESCO DE GIOVANNI ◽

ALESSIO RUSSO

Keyword(s):

Finite Group ◽

Finite Groups ◽

Upper Bound ◽

Nonnegative Integer ◽

Commutator Subgroup ◽

Study Groups ◽

Normal Closure ◽

Normal Length ◽

Sylow Tower

Let $k$ be a nonnegative integer. A subgroup $X$ of a group $G$ has normal length $k$ in $G$ if all chains between $X$ and its normal closure $X^{G}$ have length at most $k$, and $k$ is the length of at least one of these chains. The group $G$ is said to have finite normal length if there is a finite upper bound for the normal lengths of its subgroups. The aim of this paper is to study groups of finite normal length. Among other results, it is proved that if all subgroups of a locally (soluble-by-finite) group $G$ have finite normal length in $G$, then the commutator subgroup $G^{\prime }$ is finite and so $G$ has finite normal length. Special attention is given to the structure of groups of normal length $2$. In particular, it is shown that finite groups with this property admit a Sylow tower.

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