A new proof of the solvable signalizer functor theorem.
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 works in Bender's approach but not in general. Furthermore there are some special problems for Fermat primes. This makes Aschbacher's proof more complex than Bender's proof.NEWLINENEWLINE In this paper now the author gives a new proof of the solvable signalizer functor theorem following Bender's idea and replacing the ZJ-theorem by results of primitive pairs he has obtained recently [\textit{P. Flavell}, J. Algebra 324, No. 4, 841-859 (2010; Zbl 1209.20020)]. For the Fermat primes he follows Aschbacher's proof.
scientific article; zbMATH DE number 6334755
scientific article