New characteristic subgroups for \(p\)-stable finite groups and fusion systems (Q6597518)

Let \(G\) be a finite group, \(p\) an odd prime, \(P\) a Sylow \(p\)-subgroup of \(G\) and \(J(P)\) the Thompson subgroup of \(P\). The celebrated ZJ-theorem of \textit{G. Glauberman} ([Can. J. Math. 20, 1101--1135 (1968; Zbl 0164.02202)]) asserts that if \(G\) is \(p\)-stable and \(C_{G}(O_{p}(G)) \leq O_{p}(G)\), then \(Z(J(P))\) is a characteristic subgroup of \(G\).\N\NIn the paper under review the authors define a new characteristic subgroup \(\mathsf{Q}_{(e)}(P)\) of \(P\) and they prove that \(\mathsf{Q}_{(e)}(P)\) is also a characteristic subgroup of a \(p\)-stable group \(G\) having \(P\) as a Sylow \(p\)-subgroup. Furthermore they prove that \(\mathsf{Q}_{(e)}(P)\) is a normal subgroup of any \(p\)-stable fusion system over \(P\).