What question did this study set out to answer?

This research aims to explore the structure of finite groups with solvable Hall sigma-basis, focusing on subgroup conditions.

April 3, 2026

Finite Groups with a Solvable Hall -Basis

Key Points

This research aims to explore the structure of finite groups with solvable Hall sigma-basis, focusing on subgroup conditions.
Defined Hall sigma-set and its properties for finite groups.
Analyzed the existence of complete Hall sigma-sets in the groups.
Investigated conditions for subgroups to be permutable.
Established structural properties of finite groups with solvable Hall sigma-basis.
Demonstrated that permutability of subgroups influences group structure.
Characterized conditions under which all subgroups in Hall sigma-sets are permutable.

Abstract

Let = \ᵢ i I \ be a partition of the set P of all primes, and let G be a finite group. A set H of subgroups of G is called a complete Hall -set of G if every subgroup in H is a ᵢ -Hall subgroup of G for every i I and H contains exactly one ᵢ -Hall subgroup for every i such that ᵢ (G). In this paper, we study the structure of the group G ₈ ₈D㶁 (S) under the condition that all subgroups in every complete Hall -set of the group G are permutable.

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