Finite groups in which maximal subgroups of Sylow \(p\)-subgroups are \(\mathcal{M} \)-permutable (Q6636298)

Let \(G\) be a finite group and \(p\) a prime divisor of its order. Let \(P\) be a non-trivial \(p\)-subgroup of \(G\), with minimal number of generators equal to \(d\). \(P\) is said to be \(\mathcal{M}\)-permutable in \(G\) if there are \(B \le G\) and a set \(\{ P_{1}, \dots, P_{d} \}\) of maximal subgroups of \(P\) such that the intersection of the \(P_{i}\) equals the Frattini subgroup of \(P\) and we have that \(G = P B\) and that \(P_{i} B = B P_{i}\) is a proper subgroup of \(G\) for all \(i\). A composition factor of \(G\) is said to be a pd-composition factor if its order is divisible by \(p\).\N\NTheorem~A of the paper under review states the following. Let \(P\) be a Sylow \(p\)-subgroup of the finite group \(G\), for a prime divisor \(p\) of the order of \(G\). If every maximal subgroup of \(P\) is \(\mathcal{M}\)-permutable in \(G\), then the non-abelian pd-composition factors belong to a certain list, which includes some projective special linear groups for primes of a special form, two Mathieu groups for particular primes and the alternating group of degree \(p > 5\). Several results further explore other consequences of the \(\mathcal{M}\)-permutability of the maximal subgroup of \(P\).