On the Solubilizer of an Element in a Finite Group

Abstract

AbstractThe solubility graph $$\Gamma _S(G)$$ Γ S ( G ) associated with a finite group G is a simple graph whose vertices are the elements of G,  and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this paper, we focus on the set of neighbors of a vertex x which we call the solubilizer of x in G,  $$\mathrm {Sol}_G(x),$$ Sol G ( x ) , investigating both arithmetic and structural properties of this set.