SYLOW MULTIPLICITIES IN FINITE GROUPS

Let G be a finite group and let P = P-1, . . . , P-m be a sequence of Sylow p(i)-subgroups of G, where p(1), . . . , p(m) are the distinct prime divisors of vertical bar G vertical bar. The Sylow multiplicity of g is an element of G in P is the number of distinct factorizations g = g(1) , . . . , g(m) such that g(i) is an element of 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.