Supersoluble Groups and Its Products

Two subgroups A and B of a group G are called permutable if every subgroup X of A is permutable with every subgroup Y of B, i.e., XYis a subgroup of G. In this case, if G=AB we say that G is the permutable product of the subgroups A and B. In this paper we check the permutable product of supersoluble subgroups. And the end, we obtain sufficient conditions for permutable products of finite supersoluble groups to be supersoluble.

On a product of two formational tcc-subgroups

A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G=AT and for any X≤A and Y≤T there exists an element u∈⟨X,Y⟩ such that XYu≤G. The notation H≤G means that H is a subgroup of a group G. In this paper we consider a group G=AB such that A and B are tcc-subgroups in G. We prove that G belongs to F, when A and B belong to F and F is a saturated formation of soluble groups such that U⊆F. Here U is the formation of all supersoluble groups.

Theoretical Researches about u-Maximal Subgroups and Its Applications in Charactering IntuG

Let G be a finite group and u be the class of all finite supersoluble groups. A supersoluble subgroup U of G is called u-maximal in G if for any supersoluble subgroup V of G containing U, V=U. Moreover, IntuG is the intersection of all u-maximal subgroups of G. This paper obtains some new criteria on IntuG, by assuming that some subgroups of G are either Φ-I-supplemented or Φ-I-embedded in G. Here, a subgroup H of G is called Φ-I-supplemented in G if there exists a subnormal subgroup T of G such that G=HT and H∩THG/HG≤ΦH/HGIntuG and Φ-I-embedded in G if there exists a S-quasinormal subgroup T of G such that HT is S-quasinormal in G and H∩THG/HG≤ΦH/HGIntuG.

Finite groups with given sets of 𝔉-subnormal subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

ON A THEOREM OF KANG AND LIU ON FACTORISED GROUPS

Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.

On mutually permutable products of generalized supersoluble subgroups of finite groups

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.