Let $\Gamma_G $ denote a graph associated with a group $ G $. A compelling question
about finite groups asks whether or not a finite group $ H $ must be nilpotent provided …

Let G be a finite group, and let x be an element of G. Denote by Sol G ( x ) \documentclass[12pt]{minimal}
\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} …

Given a finite group G, the solubilizer of an element x, denoted by Sol G (x), is the set of all
elements y such that〈 x, y〉 is a soluble subgroup of G. In this paper, we provide a …

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic
graph) associated with a finite group. In particular, we characterise finite p-groups with the …

Let $ G $ be a finite insoluble group with soluble radical $ R (G) $. The solubility graph
$\Gamma_ {\rm S}(G) $ of $ G $ is a simple graph whose vertices are the elements of …

Let G be a finite group and x be an element of G. Let SolG (x) denote the set of all y∈ G such
that⟨ x, y⟩ is soluble. We find the structure G when SolG (x) is meta-cyclic maximal 2 …