Number of Sylow subgroups and \(p\)-nilpotence of finite groups
Let \(G\) be a finite group and let \(\pi(G)\) be the set of primes dividing the order of \(G\). The classical theorems of Sylow assert that the number \(n_p(G)\) of Sylow \(p\)-subgroups is congruent to 1 modulo \(p\), which is equal to the index \(|G:N_G(P)|\) for some Sylow \(p\)-subgroup \(P\). A finite group \(G\) is said to be \(p\)-nilpotent if \(G\) has a normal \(p\)-complement for some \(P\in\text{Syl}_p(G)\), the set of Sylow \(p\)-subgroups of \(G\). It is proved that if \(p\neq 3\) is a prime then \(G\) is \(p\)-nilpotent if and only if \((p,n_r(G))=1\) for every \(r\in\pi(G)\) and if \(G\) does not have a composition factor isomorphic with \(U_3(2^q)\), \(q\) even not divisible by 3, then \(G\) is \(p\)-nilpotent if and only if \((p,n_r(G))=1\) for every \(r\in\pi(G)\).
scientific article; zbMATH DE number 1211655
scientific article