Index graphs of finite permutation groups

Haval Mohammed Salih (Soran University)

Indonesian Journal of Combinatorics
Vol 6, No 2 (2022)

Let G be a subgroup of Sn. 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 Γind(G). Its vertex set the set of all conjugacy classes of G and two distinct vertices x ∈ Cx and y ∈ Cy are adjacent if Gcd(ind x, ind y) 6 ≠ 1. We study some properties of this graph for the symmetric groups Sn, the alternating group An, the cyclic group Cn, the dihedral group D2n and the generalized quaternain group Q4n. In particular, we are interested in the connectedness of them.