Random Subgraphs of Cayley Graphs overp-Groups

The subject of this paper is the size of the largest component in random subgraphs of Cayley graphs, X_n, taken over a class of p -groups, G_n. G_nconsists of p -groups, G_n, with the following properties: (i)G_n / Φ(G_n)  = F_pⁿ, where Φ(G_n) is the Frattini subgroup and (ii) | G_n|  ≤ n^Kn, where K is some positive constant. We consider Cayley graphs X_n = Γ(G_n, S_n′), where S_n′ = S_n  S_n ^− 1, and S_nis a minimal G_n-generating set. By selecting G_n-elements with the independent probability λ_nwe induce random subgraphs of X_n. Our main result is, that there exists a positive constant c >  0 such that for λ_n = c ln(| S_n′ |) / | S_n′ | the largest component of random induced subgraphs of X_ncontains almost all vertices.