Sylow multiplicities in finite groups

Let $G$ be a finite group and let $\mathcal{P}=P_{۱},\ldots,P_{m}$ be a sequence‎ ‎of Sylow $p_{i}$-subgroups of $G$‎, ‎where $p_{۱},\ldots,p_{m}$ are the distinct‎ ‎prime divisors of $\left\vert G\right\vert $‎. ‎The Sylow multiplicity of $g\in‎ ‎G$ in $\mathcal{P}$ is the number of distinct factorizations $g=g_{۱}\cdots‎ ‎g_{m}$ 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.

Keywords:

‎Sylow sequences‎ , ‎Sylow multiplicities‎ , ‎Solvable radical‎ , ‎Solvable residual