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, Xn, taken over a class of p -groups, Gn. Gnconsists of p -groups, Gn, with the following properties: (i)Gn / Φ(Gn) ∼ = Fpn, where Φ(Gn) is the Frattini subgroup and (ii) | Gn| ≤ nKn, where K is some positive constant. We consider Cayley graphs Xn = Γ(Gn, Sn′), where Sn′ = Sn ∪ Sn − 1, and Snis a minimal Gn-generating set. By selecting Gn-elements with the independent probability λnwe induce random subgraphs of Xn. Our main result is, that there exists a positive constant c > 0 such that for λn = c ln(| Sn′ |) / | Sn′ | the largest component of random induced subgraphs of Xncontains almost all vertices.