Finite groups with given systems of generalised σ-permutable subgroups

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

On groups with conjugate-permutable subgroups

According to Foguel, a subgroup [Formula: see text] of a group [Formula: see text] is called conjugate-permutable if [Formula: see text] for every [Formula: see text]. Mingyao Xu and Qinhai Zhang studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three questions about them. We gave the answers to these questions. In particular, every group of exponent 3 is an ECP-group, there exist non-regular ECP-[Formula: see text]-groups and the class of all finite ECP-groups is neither a formation nor a variety.

ON CERTAIN PRODUCTS OF PERMUTABLE SUBGROUPS

In this paper, we study the structure of finite groups $G=AB$ which are a weakly mutually $sn$ -permutable product of the subgroups A and B, that is, A permutes with every subnormal subgroup of B containing $A \cap B$ and B permutes with every subnormal subgroup of A containing $A \cap B$ . We obtain generalisations of known results on mutually $sn$ -permutable products.

Groups whose non-permutable subgroups are metaquasihamiltonian

If {\mathfrak{X}} is a class of groups, define a sequence of group classes {\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots,\mathfrak{X}_{k},\ldots} by putting {\mathfrak{X}_{1}=\mathfrak{X}} and choosing {\mathfrak{X}_{k+1}} as the class of all groups whose non-permutable subgroups belong to {\mathfrak{X}_{k}}. In particular, if {\mathfrak{A}} is the class of abelian groups, {\mathfrak{A}_{2}} is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of {\mathfrak{X}_{k}}-groups, with special emphasis on the case {\mathfrak{X}=\mathfrak{A}}. Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to {\mathfrak{A}_{k}} for some positive integer k.

Groups with restricted non-permutable subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be permutable if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] and the group [Formula: see text] is called metaquasihamiltonian if all subgroups of [Formula: see text] are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group [Formula: see text] is soluble with derived length at most [Formula: see text] and contains a finite normal subgroup [Formula: see text] such that all subgroups of the factor [Formula: see text] are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.