Index graphs of finite permutation groups

Abstract

Let G be a subgroup of S_n. For x ∈ G, the index of x in G is denoted by ind x is the minimal number of 2-cycles needed to express x as a product. In this paper, we define a new kind of graph on G, namely the index graph and denoted by Γ^ind(G). Its vertex set the set of all conjugacy classes of G and two distinct vertices x ∈ C_x and y ∈ C_y are adjacent if Gcd(ind x, ind y) 6 ≠ 1. We study some properties of this graph for the symmetric groups S_n, the alternating group A_n, the cyclic group C_n, the dihedral group D_2n and the generalized quaternain group Q_4n. In particular, we are interested in the connectedness of them.