Fixed points of the complements of Frobenius groups of automorphisms. (Q606062)

Let \(G\) be a finite group, let \(A\leqslant\Aut(G)\) and let \(N\) be a normal \(A\)-invariant subgroup of \(G\). The author is interested in finding sufficient conditions for the equality \(G_{G/N}(A)=C_G(A)N/N\) to hold. This is shown to happen (Theorem 1) when \(A=KC\) is a Frobenius group with kernel \(K\) and complement \(C\) such that \((|N|,|K|)=1\) and \(C_N(K)=1\). The author mentions that P. Shumyatsky noticed that the coprimeness condition can be dropped. As a direct consequence, it is shown in Theorem 2 that if, moreover, the group \(G\) in Theorem 1 is nilpotent, the fixed point subring of the action induced by \(A\) on the associated Lie ring \(L(G)\) can be nicely described as a direct sum of the fixed point subrings induced by \(A\) on the factors of the lower central series.