Arithmetic graphs and the products of finite groups

Abstract

The Hawkes graph ΓH(G) of G is the directed graph whose vertex set coincides with π(G) and it has the edge (p, q) whenever q ∈ π(G/Op′,p(G)). The Sylow graph Γs(G) of G is the directed graph with vertex set π(G) and (p, q) is an edge of Γs(G) whenever q ∈ π(NG(P)/PCG(P)) for some Sylow p-subgroup P of G. The N-critical graph ΓNc(G) of a group G the directed graph whose vertex set coincides with π(G) such that (p, q) is an edge of ΓNc(G) whenever G contains a Schmidt (p, q)-subgroup, i.e. a Schmidt {p, q}-subgroup with a normal Sylow p-subgroup. In the paper the Hawkes, the Sylow and the N-critical graphs of the products of totally permutable, mutually permutable and N-connected subgroups are studied.