On the Solubilizer of an Element in a Finite Group

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