# Sylow multiplicities in finite groups

Document Type : Ischia Group Theory 2016

Author

Italy

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 $\left\vert G\right\vert$‎. ‎The Sylow multiplicity of $g\in‎ ‎G$ 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.

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