A note on \(\pi\)-nilpotent groups (Q1864685)

The author generalizes the celebrated Zassenhaus-Schur theorem by proving that if a finite group \(G\) has a normal subgroup \(K\) and a subgroup \(H\) contained in \(K\) such that \((|G:K|,|K:H|)=1\), then \(K\) has a relative complement over \(H\) in \(G\) (i.e., a subgroup \(L\) of \(G\) such that \(G=KL\) and \(K\cap L=H\)) if and only if \(G=KN_G(H)\). Moreover, any two relative complements of \(K\) over \(H\) in \(G\) are conjugated. He applies this theorem in the study of \(\pi\)-nilpotent groups defined as follows. Let \(\pi\) be a non-empty set of prime numbers. A finite group \(G\) is said to be \(\pi\)-nilpotent over its subgroup \(H\) if \(G\) has a normal subgroup \(K\) containing \(H\) such that \(G=KN_G(H)\), \((|G:K|,|K:H|)=1\) and \(G/K\) is a nilpotent \(\pi\)-group with \(\pi(G/K)=\pi\). In case \(H=\{1\}\) \(G\) is called a \(\pi\)-nilpotent group. The author proves that a finite group \(G\) is \(\pi\)-nilpotent over its subgroup \(H\) if and only if for every \(p\in\pi\) the normalizer \(N_G(H)\) contains a Sylow \(p\)-subgroup \(P\) and for every pair of elements \(x,y\in P\) that are conjugated in \(G\) there exists \(t\in P\) such that \(xH=(yH)^t\). As a consequence, a finite group \(G\) is \(\pi\)-nilpotent if and only if for every \(p\in\pi\) and for every Sylow \(p\)-subgroup \(P\) of \(G\) every pair of elements of \(P\) that are conjugated in \(G\) are conjugated also in \(P\).