Sylow multiplicities in finite groups

Abstract

Let $G$ be a finite group and let $mathcal{P}=P_{1},ldots,P_{m}$ be a sequence‎ ‎of Sylow $p_{i}$-subgroups of $G$‎, ‎where $p_{1},ldots,p_{m}$ are the distinct‎ ‎prime divisors of $leftvert Grightvert $‎. ‎The Sylow multiplicity of $gin‎ ‎G$ in $mathcal{P}$ is the number of distinct factorizations $g=g_{1}cdots‎ ‎g_{m}$ such that $g_{i}in P_{i}$‎. ‎We review properties of the solvable‎ ‎radical and the solvable residual of $G$ which are formulated in terms of‎ ‎Sylow multiplicities‎, ‎and discuss some related open questions‎.