Arithmetic graphs and factorized finite groups

The Hawkes graph of a group G is the directed graph with vertex set pi(G) that has an edge (p, q) whenever q is an element of pi (G/O-p',O-p(G)). The Sylow graph Gamma(s)(G) of a group G is the directed graph with vertex set pi(G) that has an edge (p, q) whenever q is an element of pi(N-G(P)/PCG(P)) for some Sylow p-subgroup P of G. The N-critical graph Gamma(Nc) (G) of a group G is the directed graph with vertex set pi(G) that has an edge (p, q) whenever G contains a Schmidt (p, q)-subgroup, i.e., a Schmidt {p, q}-subgroup with a normal Sylow p-subgroup. The paper studies the Hawkes, Sylow, and N-critical graphs of products of totally permutable, mutually permutable, and n-connected subgroups.