Abstract
The solubility graph Gamma _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), investigating both arithmetic and structural properties of this set.
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