On \(\pi\)-separable groups (Q1125879)

Let \(G\) be a finite group and let \(I(A)=\{B\leq G\mid A\leq N_G(B)\) and \(A\cap B=\{1\}\}\). Let \(I(A,\pi')=\{B\in I(A)\mid B\) is a \(\pi'\)-subgroup\}. Generalizations of some results from the odd order paper are obtained here. New results are extended from \(p\)-solvable groups to \(\pi\)-separable groups, where \(\pi\) is an arbitrary set of primes. In particular it is proved, that if \(H\) is a \(\pi\)-Hall subgroup of a group \(G\) and \(A\trianglelefteq H\) such that \(C_H(A)\leq A\) then: (1) if \(\pi=\{p\}\) then \(I(A)\) contains only \(p'\)-subgroups; (2) if \(G\) is \(\pi\)-separable then \(I(A,\pi')\) is a lattice whose maximal element is \(O_{\pi'}(G)\).