On the cohomology of finite soluble groups.
type: Item
instance of: scholarly article
title: On the cohomology of finite soluble groups.
zbMATH Open document ID: 1335.20053
author: Derek J. S. Robinson
published in: Archiv der Mathematik
publication date: 2015-09-01T00:00:00Z
review text: In this paper, a theorem of \textit{C. Parker} and \textit{P. Rowley} [Bull. Lond. Math. Soc. 42, No. 3, 417-419 (2010; Zbl 1193.20019)] is revisited and two vanishing theorems (Theorems 1 and 3) for group cohomology at dimensions 1 and 2 are proved. The author remarks that the vanishing theorem at dimension 1 is what is behind the Parker-Rowley theorem. Finally, some applications of these theorems are given. If \(\Omega\) is a set of unary operations on a group \(G\), then \(\mathrm{comp}_\Omega(G)\) denotes the set of isomorphism classes of \(\Omega\)-composition factors of \(G\). The Parker-Rowley theorem states the following: Suppose that \(N\) is a nilpotent normal subgroup of a finite solvable group \(G\) such that \(\mathrm{comp}_G(G/N)\cap\mathrm{comp}_G(N)\) is empty. If \(H\) and \(K\) are subgroups of \(G\) such that \(HN=KN=G\) and \(H\cap N=K\cap N\), then \(H\) and \(K\) are conjugate in \(G\). Motivated by this theorem, the following cohomological vanishing theorem is proved: Theorem 1. Let \(Q\) be a finite group and \(A\) be a finite \(Q\)-module. Assume that \(Q\) is \(p\)-solvable for all the primes \(p\) dividing the order of \(A\), and that \(\mathrm{comp}_Q(Q)\cap\mathrm{comp}_Q(A)\) is empty. Then \(H^1(Q,A)=0\). Using this theorem, a generalization of the Parker-Rowley theorem is also given (Theorem 2). Then in Section 3, a similar vanishing result for group cohomology at dimension 2 is proved (Theorem 3). The assumptions of Theorem 3 are too technical to state here. The paper ends with some applications of these theorems to the splitting of group extensions and to the structure of automorphism groups of finite solvable groups with nilpotent length at most \(3\).
reviewed by: Ergün Yalçın
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6477366
zbMATH Keywords: cohomology of groups
zbMATH Keywords: finite soluble groups
MaRDI profile type: Publication
full work available at URL: s00013-015-0790-1
OpenAlex ID: W978779772
cites work: ON CONJUGACY OF SUPPLEMENTS OF NORMAL SUBGROUPS OF FINITE GROUPS
cites work: On finite complete groups
cites work: Q3125202
cites work: The Wells exact sequence for the automorphism group of a group extension.
cites work: A note on conjugacy of supplements in finite soluble groups
cites work: Automorphisms of Group Extensions
DOI: 10.1007/S00013-015-0790-1
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description: scientific article; zbMATH DE number 6477366