On the existence of Hall subgroups
type: Item
instance of: scholarly article
title: On the existence of Hall subgroups
zbMATH Open document ID: 0588.20018
DOI: 10.1016/0021-8693(86)90012-8
author: Fletcher Gross
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: This paper is a happy reminder of how beautiful finite group theory can still be. It is concerned with theorems like Sylow's and the conditions \(E_{\pi}\), \(C_{\pi}\), \(D_{\pi}\) introduced by Hall; the group G satisfies \(E_{\pi}\) if G possesses Hall \(\pi\)-subgroups, \(C_{\pi}\) if G has a unique class of them and \(D_{\pi}\) if there is a Hall \(\pi\)- subgroup of G which contains a conjugate of any \(\pi\)-subgroup of G. Assuming the Schreier hypothesis, (as is done throughout the paper), Chunikhin showed that if N is normal in G and N, G/N both satisfy \(C_{\pi}\), then G satisfies \(C_{\pi}.\) But the corresponding statement for \(E_{\pi}\) is false. It is proved (Theorem 3.4) that if N and G/N both satisfy \(E_{\pi}\) but G does not, then G has a non-Abelian composition factor H/K such that H/K does satisfy \(E_{\pi}\) but \(Aut_ G(H/K)\) does not. Here \(Aut_ G(H/K)\) means the group of automorphisms of H/K induced by \(N_ G(H)\cap N_ G(K)\), so it is dependent on H and K, not merely on the composition factor H/K. In fact an example is given of a group G with two composition series, where all the factors X of one fulfil the condition that \(Aut_ GX\) satisfies \(E_{\pi}\) but some of those of the other do not. Let us start, then, with the condition (d) that \(Aut_ G(H/K)\) satisfies \(E_{\pi}\) for all composition factors H/K of all composition series of G. Let \(1=G_ 0<G_ 1<...<G_ n=G\) be a composition series which is a refinement of a chief series of G. Trivially, (d) implies the condition (c) that \(Aut_ G(G_ i/G_{i-1})\) satisfies \(E_{\pi}\) for all i. But also such a series has the property that if A is an atom of \(G_ i\) with maximal normal subgroup \(A^*\), then \(A/A^*\cong G_ i/G_{i- 1}\) and \(Aut_ G(A/A^*)\cong Aut_ G(G_ i/G_{i-1})\) (Lemma 2.5). (An atom of G, as defined by Wielandt, is a perfect subnormal subgroup with a unique maximal normal subgroup.) So (c) implies the condition (b) that \(Aut_ G(A/A^*)\) satisfies \(E_{\pi}\) for eve…
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 3944134
zbMATH Keywords: Hall \(\pi \) -subgroups
zbMATH Keywords: Schreier hypothesis
zbMATH Keywords: non-Abelian composition factor
zbMATH Keywords: group of automorphisms
zbMATH Keywords: composition series
zbMATH Keywords: composition factors
zbMATH Keywords: chief series
zbMATH Keywords: maximal normal subgroup
zbMATH Keywords: atom
zbMATH Keywords: perfect subnormal subgroup
reviewed by: Norman Blackburn
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90012-8
OpenAlex ID: W1970156739
cites work: On normal subgroups which are direct products
cites work: Theorems Like Sylow's
cites work: Zum Satz von Sylow
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description: scientific article; zbMATH DE number 3944134