On the Hall \(D_ \pi\)-property for finite groups
type: Item
instance of: scholarly article
title: On the Hall \(D_ \pi\)-property for finite groups
zbMATH Open document ID: 0880.20012
DOI: 10.1007/BF02674906
published in: Siberian Mathematical Journal
publication date: 1998-01-05T00:00:00Z
review text: A finite group is said to satisfy the property \(E_\pi\), if it possesses a Hall \(\pi\)-subgroup; \(C_\pi\), if it satisfies the property \(E_\pi\) and all its Hall \(\pi\)-subgroups are conjugate; \(D_\pi\), if it satisfies the property \(C_\pi\) and every \(\pi\)-subgroup of it is contained in some Hall \(\pi\)-subgroup. The above notation was introduced by P. Hall and the properties generalize those of a Sylow \(p\)-subgroup. P. Hall also raised the following question: Is an extension of a \(D_\pi\)-group by another \(D_\pi\)-group again a \(D_\pi\)-group? There is no exhaustive answer to this question, although an affirmative answer for \(C_\pi\)-groups is well known. Let \(S\) be a normal and homomorphism-closed class of finite groups. Say that the \(D_\pi\)-conjecture is valid for \(S\) if an extension of every \(D_\pi\)-group in \(S\) by an arbitrary \(D_\pi\)-group is a \(D_\pi\)-group. Say that the \(D_\pi\)-conjecture is valid for almost every simple group in \(S\) if, for every nonabelian simple group \(P\in S\), the group \(\text{Out}(P)\) is solvable and the property \(D_\pi\) is satisfied for every group \(G\) that satisfies the following conditions: 1) \(\text{Inn}(P)\leq G\leq\Aut(P)\), with \(P\) a nonabelian simple group; 2) \(G\) possesses a normal subgroup \(A\) in \(S\) that contains \(P\) and satisfies the property \(D_\pi\). Call a class \(S\) \(D_\pi\)-normal if the property \(D_\pi\) is hereditary for normal subgroups in \(S\). The authors of the present article show that the general Hall problem can be reduced to an examination of simple groups and their automorphism groups. In particular, they prove the following result: Theorem 1. Let \(S\) be a normal and homomorphism-closed class of finite groups. If the \(D_\pi\)-conjecture is valid for almost every simple group in \(S\) then it is valid for \(S\). Theorem 2 provides sufficient conditions for \(D_\pi\)-normality. These two theorems arouse the hope that an answer to P. Hall's question …
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 1099651
zbMATH Keywords: Hall subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: finite groups
author: V. D. Mazurov
author: D. O. Revin
MaRDI profile type: Publication
cites work: Theorems Like Sylow's
cites work: Q4870316
cites work: Conjugacy of Odd order Hall Subgroups
cites work: Hall subgroups of the symmetric groups
cites work: Solvability of groups of odd order
cites work: The characterization of finite groups with abelian Sylow 2-subgroups
cites work: Structure of Ree groups
cites work: Q3684278
cites work: On a Conjecture of Philip Hall
description: scientific article; zbMATH DE number 1099651