A new proof of the solvable signalizer functor theorem.
type: Item
instance of: scholarly article
title: A new proof of the solvable signalizer functor theorem.
zbMATH Open document ID: 1312.20020
published in: Journal of Algebra
publication date: 2014-08-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1609.08311
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 6334755
zbMATH Keywords: finite simple groups
zbMATH Keywords: finite solvable groups
zbMATH Keywords: solvable signalizer functors
author: Paul Flavell
MaRDI profile type: Publication
OpenAlex ID: W2068504034
arXiv ID: 1609.08311
cites work: Q3708174
cites work: Goldschmidt's 2-signalizer functor theorem
cites work: Primitive pairs of \(p\)-solvable groups.
cites work: On Solvable Signalizer Functors in Finite Groups
cites work: Solvable signalizer functors on finite groups
cites work: 2-signalizer functors on finite groups
cites work: Q3941595
cites work: Q4321756
cites work: Q4857335
cites work: The theory of finite groups. An introduction.
cites work: Q3712472
DOI: 10.1016/J.JALGEBRA.2012.12.009
review text: In this paper the author proves: Let \(G\) be a finite group, \(A\) an elementary abelian \(r\)-subgroup of \(G\) with rank \(m(A)\geq r\) and \(\Theta\) be a solvable \(A\)-signalizer functor on \(G\), then \(G\) is solvable complete. Here an \(A\)-signalizer functor is a mapping attaching each \(a\in A^\#\) an \(A\)-invariant \(r'\)-subgroup \(\Theta(C_G(a))\) of \(C_G(a)\) such that \(\Theta(C_G(a))\cap C_G(b)\leq\Theta(C_G(b))\), for all \(a,b\in A^\#\). The functor is called solvable if \(\Theta(C_G(a))\) is solvable and is called solvable complete if there is a solvable normal \(r'\)-subgroup \(M\) of \(G\) such that \(\Theta(C_G(a))=C_M(a)\) for all \(a\in A^\#\). Signalizer functors play an important role in the classification of the finite simple groups, in particular to get control over \(O(C_G(i))\) for \(i\) an involution in \(G\).NEWLINENEWLINE Signalizer functors in this sense were introduced by \textit{D. Gorenstein} [J. Algebra 11, 243-277 (1969; Zbl 0214.27902)] generalizing some ideas of J. Thompson in the \(N\)-group paper. In [J. Algebra 14, 350-372 (1970; Zbl 0202.02402)] \textit{D. Gorenstein} proved the solvable signalizer functor theorem for \(r=2\) and \(m(A)\geq 5\). This was improved by \textit{D. M. Goldschmidt} to \(r\) odd and \(m(A)\geq 4\) [in J. Algebra 21, 137-148 (1972; Zbl 0253.20032)] and \(r=2\), \(m(A)\geq 3\) [in J. Algebra 21, 321-340 (1972; Zbl 0253.20033)]. In the generality as above the solvable signalizer functor theorem has been proved by \textit{G. Glauberman} [Proc. Lond. Math. Soc., III. Ser. 33, 1-27 (1976; Zbl 0342.20008)]. For \(r=2\) there is a short proof due to \textit{H. Bender} [Isr. J. Math. 22, 208-213 (1976; Zbl 0325.20016)]. \textit{M. Aschbacher} in the first edition of his book [Finite group theory. Cambridge Studies in Advanced Mathematics, 10. Cambridge: Cambridge University Press (1986; Zbl 0583.20001)] gave a proof generalizing Bender's ideas. The main problem is the use of the ZJ-Theorem, which work…
description: scientific article; zbMATH DE number 6334755