Index graphs of finite permutation groups

Abstract

Let <em>G</em> be a subgroup of <em>S</em><sub>n</sub>. For <em>x ∈ G</em>, the index of <em>x</em> in <em>G</em> is denoted by <em>ind x</em> is the minimal number of 2-cycles needed to express <em>x</em> as a product. In this paper, we define a new kind of graph on <em>G</em>, namely the index graph and denoted by <em>Γ</em><sup>ind</sup><em>(G)</em>. Its vertex set the set of all conjugacy classes of <em>G</em> and two distinct vertices <em>x ∈ C</em><sub>x</sub> and <em>y ∈ C</em><sub>y</sub> are adjacent if <em>Gcd(ind x, ind y) 6 ≠ 1</em>. We study some properties of this graph for the symmetric groups <em>S</em><sub>n</sub>, the alternating group <em>A</em><sub>n</sub>, the cyclic group <em>C</em><sub>n</sub>, the dihedral group <em>D</em><sub>2n</sub> and the generalized quaternain group <em>Q</em><sub>4n</sub>. In particular, we are interested in the connectedness of them.