sylow multiplicities in finite groups

Let gg be a finite group and let p=p1,…,pm be a sequence‎ ‎of sylow pipi-subgroups of g‎, ‎where p1,…,pm are the distinct‎ ‎prime divisors of |g|. ‎the sylow multiplicity of g∈‎‎g in pp is the number of distinct factorizations g=g1⋯‎‎gm such that gi∈pi. ‎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.