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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