Let σ = {σ_{i} | *i* ∈ I} be a partition of the set of all primes ℙ and *G* a finite group. Suppose σ(*G*) = {σ_{i} | σ_{i} ∩ π(*G*) ≠ = ∅}. A set ℋ of subgroups of *G* is called a complete Hall σ-set of *G* if every nontrivial member of ℋ is a σ_{i}-subgroup of *G* for some *i* ∈ *I* and ℋ contains exactly one Hall σ_{i}-subgroup of *G* for every *i* such that σ_{i} ∈ σ(*G*). A group *G* is σ-full if *G* possesses a complete Hall σ-set. A complete Hall σ-set ℋ of *G* is called *a* σ-basis of *G* if every two subgroups *A, B* ∈ ℋ are permutable, i.e., *AB* = *BA*. In this paper, we study properties of finite groups having a σ-basis. It is proved that if *G* has a σ-basis, then *G* is generalized σ-soluble, i.e, |σ(*H*/*K*)| ≤ 2 for every chief factor H/K of *G*. Moreover, it is shown that every complete Hall σ-set of a σ-full group *G* forms a σ-basis of *G* iff *G* is generalized σ-soluble, and for the automorphism group *G*/*C*_{G}(H/K) induced by *G* on any its chief factor H/K, we have |σ(*G/C*_{G}(H/K))| ≤ 2 and also σ(H/K) ⊆ σ(*G/C*_{G}(H/K)) in the case |σ(*G/C*_{G}(H/K))| = 2.

## Keywords

finite group Hall subgroup σ-soluble subgroup σ-basis generalized σ-soluble group## Preview

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