THE ISOTROPY GRAPH OF FINITE NON-ABELIAN GROUPS

Abstract

Let G be a finite non-abelian group and let be a set of elements of G. Let A be the set of commuting elements in , i.e A = {v ∈ : vg = gv,g ∈ G}. In this paper, we extend the work on centralizer graph by defining a new graph called the isotropy graph, denoted as iso. The vertices of this graph are proper isotropy groups. In other words, |V ( iso)| = StabG()−A, where StabG() is the number of stabilizers under group action of G on and A is the number of improper isotropy groups under group action on a set. Two vertices are connected by an edge if their cardinalities are identical.