On weakly τ-quasinormal subgroups of finite groups

Rendiconti del Seminario Matematico della Università di Padova, 2011

For each prime p and positive integer n, Berger and Gross have defined a finite p-group G HX, where H is a core-free quasinormal subgroup of exponent p nÀ1 and X is a cyclic subgroup of order p n. These groups are universal in the sense that any other finite p-group, with a similar factorisation into subgroups with the same properties, embeds in G. In our search for quasinormal subgroups of finite p-groups, we have discovered that these groups G have remarkably few of them. Indeed when p is odd, those lying in H can have exponent only p, p nÀ2 or p nÀ1. Those of exponent p are nested and they all lie in each of those of exponent p nÀ2 and p nÀ1 .

Journal of Algebra, 2008

The following concept is introduced: a subgroup H of the group G is said to be SS-quasinormal (Supplement-Sylow-quasinormal) in G if H possesses a supplement B such that H permutes with every Sylow subgroup of B. Groups with certain SS-quasinormal subgroups of prime power order are studied. For example, fix a prime divisor p of |G| and a Sylow p-subgroup P of G, let d be the smallest generator number of P and M d (P ) denote a family of maximal subgroups P 1 , . . . , P d of P satisfying d i=1 (P i ) = Φ(P ), the Frattini subgroup of P . Assume that the group G is p-solvable and every member of some fixed M d (P ) is SS-quasinormal in G, then G is p-supersolvable.

Communications in Algebra®, 2008

IRJET, 2020

A subgroup H of a group G is permutable subgroup of G if for all subgroups S of G the following condition holds SH = HS < S,H >. A subgroup H is S-quasinormal in G if it permutes with every Sylow subgroup of G. In this article we study the influence of S-quasinormality of subgroups of some subgroups of G on the super-solvability of G.

2004

Mathematical Notes, 2014

A subgroup H of a group G is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded in G if for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.

A subgroup H of a group G is called inert if for each $g\in G$ the index of $H\cap H^g$ in $H$ is finite. We show that for a subnormal subgroup $H$ this is equivalent to being strongly inert, that is for each $g\in G$ the index of $H$ in the join $\langle H,H^g\rangle$ is finite for all $g\in G$. Then we give a classification of soluble-by-finite groups $G$ in which subnormal subgroups are inert in the cases $G$ has no nontrivial periodic normal subgroups or $G$ is finitely generated.

Rendiconti del Seminario Matematico della Università di Padova, 2010

Let H be a subgroup of a finite group G. We say that H is t-quasinormal in G if HP PH for all Sylow p-subgroups P of G such that (jHj; p) 1 and (jHj; jP G j) T 1. In this article, finite groups in which t-quasinormality is a transitive relation are described.

Monatshefte für Mathematik, 2005

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.

Communications in Algebra, 2017

Let Z be a complete set of Sylow subgroups of a nite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called Z-S-semipermutable if H permutes with every Sylow p-subgroup of G in Z for all p / ∈ π(H); H is said to be Z-S-seminormal if it is normalized by every Sylow p-subgroup of G in Z for all p / ∈ π(H). The main aim of this paper is to characterize the Z-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in Z are Z-S-semipermutable in G and the Z-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in Z are Z-S-seminormal in G.