| Let Gamma be a finite group, let X be a subset of Gamma where X-1 = X and 1 ∉ X. The conjugacy graph Con(Gamma; X) has vertex set Gamma and two vertices g, h ∈ Gamma are adjacent if and only if there exists x ∈ X with g = xhx-1. Let O be a group with generating set Delta. The conjugacy number con(O; Delta) is defined as the minimum integer k ≥ 2 for which there exists a nonabelian group Gamma of order k|O| and a subset X of Gamma such that Cay(O; Delta) is isomorphic to a component of Con(Gamma; X). We call this Gamma a conjugacy group for O and Delta. We will calculate the conjugacy numbers for C6, C8 and C10 and identify possible conjugacy groups. Finally we will verify that certain groups of order 4 n cannot be conjugacy groups for C2 n. |
