Nonstabilizing graphs arising from group actions
M. Bahrami-Taghanaki, Tuval Foguel, A. R. Moghaddamfar, Irene Naomi Nakaoka, J. Schmidt · Communications in Algebra · 2024
Consider a finite group G which acts on itself such that the relation ∼ on G, which is defined by x∼y iff x is a fixed point of y under this action, is both reflexive and symmetric. Let fix(G) represent the set of fixed points of G, i.e., fix(G)={x∈G | xg=x for all g∈G}. We associate with G a new graph using G∖fix(G) as its set of vertices, connecting vertices x,y∈G∖fix(G) iff x is a fixed point of y. This graph is referred to as the stabilizing graph of G and is denoted by Δs(G). Additionally, the complement of the stabilizing graph Δs(G), denoted by ∇s(G), is named the nonstabilizing graph of G. The aim of this study is to analyze various properties of the nonstabilizing graph ∇s(G) and to explore how this graph-theoretical concept relates to the broader understanding of group actions.