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.