Finite groups with permuteral primary subgroups (Q6537092)

Let \(H\) be a subgroup of a finite group \(G\). The permutizer \(P_G(H)\) is the subgroup generated by all cyclic subgroups of \(G\) that permute with \(H\), that is \(P_G(H)=\langle x\in G \, | \, \langle x\rangle H=H\langle x\rangle \rangle\). We say that \(H\) is permuteral in \(G\) if \(P_G(H) = G\) and strongly permuteral in \(G\) if \(P_U(H) = U\) for every subgroup \(U\) of \(G\) such that \(H\leq U\leq G\). Let \(\mathbb{P}\) be the set of all primes. We say that \(H\) is \(\mathbb{P}\)-subnormal in \(G\) if there is a chain of subgroups \(H=H_0\leq H_1\leq \cdots\leq H_n=G\), such that \(|H_i : H_{i-1}|\in\mathbb{P}\cup \{1\}\) for each \(i\). A group of prime power order is called a primary group and the class of all groups with \(\mathbb{P}\)-subnormal primary cyclic subgroups is denoted by \(\mathrm{v}\mathfrak{U}\).\N\NIn the paper under review, the authors determine all non-abelian finite simple groups with a \(\mathbb{P}\)-subnormal or strongly permuteral Sylow subgroup. More precisely, they prove the following theorem:\N\NTheorem A. Let \(G\) be a non-abelian finite simple group and let \(R\) be a Sylow \(r\)-subgroup of \(G\) that is \(\mathbb{P}\)-subnormal in \(G\). Then, \(r=2\) and \(G\) is isomorphic to \(L_2(7)\), \(L_2(11)\) or \(L_2(2^m)\) where \(2^m+1\in\mathbb{P}\). If, in addition, \(R\) is strongly permuteral in \(G\), then \(G\) is isomorphic to \(L_2(7)\).They also consider primary cyclic subgroups and show the following.\N\NTheorem B. If every primary cyclic subgroup of a finite group \(G\) is strongly permuteral, then \(G\) is supersoluble.\N\NTheorem C. If every primary cyclic subgroup of a group G is \(\mathbb{P}\)-subnormal or strongly permuteral, then \(G\in\mathrm{v}\mathfrak{U}\).