Non-subnormal subgroups of groups (original) (raw)

On Groups with All Subgroups Subnormal or Soluble of Bounded Derived Length

arXiv: Group Theory, 2012

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

On groups whose subnormal subgroups are inert

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

Permutable subnormal subgroups of finite groups

Archiv der Mathematik, 2009

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT-groups. In particular, it is shown that the finite solvable PT-groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not PT-groups but all subnormal subgroups of defect two are permutable.

A note on the solvability of groups

Journal of Algebra, 2006

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

On supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal

Revista de la Unión Matemática Argentina, 2019

A finite group G is called an MSN *-group if it is supersolvable, and all maximal subgroups of the Sylow subgroups of G are subnormal in G. A group G is called a minimal non-MSN *-group if every proper subgroup of G is an MSN *-group but G itself is not. In this paper, we obtain a complete classification of minimal non-MSN *-groups.

IRJET- On Some Minimal S-Quasinormal Subgroups of Finite Groups

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.