A partial analogue of Borel's fixed point theorem for finite \(p\)-groups.
This very interesting paper deals with analogues of Borel's Fixed Point Theorem. This states that if \(G\) is a connected, solvable algebraic group that acts morphically on a non-empty complete variety, then \(V\) has a fixed point under \(G\) [p.~134 of \textit{A. Borel}, Linear algebraic groups. 2nd enlarged ed. New York: Springer-Verlag (1991; Zbl 0726.20030)]. When \(V\) is a family of subgroups of \(G\), on which \(G\) acts by conjugation, then a fixed point is a normal subgroup of \(G\). \textit{B. Kostant} used a variation of the latter situation to prove that if a unipotent complex algebraic group \(G\) has an abelian algebraic subgroup of dimension \(k\), for some \(k\), then \(G\) has a normal abelian algebraic subgroup of dimension \(k\) [Topology 3, Suppl. 2, 147-159 (1965; Zbl 0134.03504)]. \textit{J. L. Alperin} and the author of the paper under review obtained an analogue of Kostant's result for a finite \(p\)-group, and a suitable family of elementary abelian subgroups [J. Algebra 203, No. 2, 533-566 (1998; Zbl 0964.20011)]. These results were further extended by the author, allowing the subgroups to have arbitrary exponent, provided they have small nilpotence class [J. Algebra 272, No. 1, 128-153 (2004; Zbl 1048.20004)]. In the paper under review, the results of the latter paper are used to obtain further generalisations of Kostant's result, some of them without restriction on the nilpotence class of the subgroups. Among the wealth of results, some of which have inevitably rather technical statements, Corollary 1 states that if \(S\) is a finite \(p\)-group which is the product of normal subgroups, each of class at most \(p-1\), and if \(B\) is a subgroup of \(S\), then there is a normal subgroup \(B^*\) of \(S\) which has the same order as \(B\), and nilpotence class at most that of \(B\). There is such a subgroup \(B^*\) within the normal closure of \(B\) in \(S\).
scientific article; zbMATH DE number 6537420
scientific article