Some questions on quasinilpotent groups and related classes. (Q1394492)

Let \(G\) be a finite group. As usual, \(F^*(G)\) denotes the generalised Fitting subgroup, i.e. the product of the Fitting subgroup and the semisimple radical of \(G\). A group is called quasinilpotent if it coincides with its generalised Fitting subgroup. If \(X\) is a subgroup of \(G\) and \(Y\) is a quasinilpotent subgroup of \(G\) such that \(XF^*(G)\) is quasinilpotent, \(X\) is subnormal in \(XF^*(G)\) and \(Y\) normalises \(X\), then \(YF^*(N_G(X))\) is quasinilpotent if and only if \(YF^*(G)\) is quasinilpotent (Theorem 6). Further, \(F^*(G)\) controls its own \(G\)-fusion (i.e. any two elements of \(F^*(G)\) that are conjugate in \(G\) are conjugate in \(F^*(G)\)) only if \(G\) is quasinilpotent (Theorem 10).