These notes concern aspects of various graphs whose vertex set is a group $ G $ and
whose edges reflect group structure in some way (so that they are invariant under the action …

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and
Kantor proved that every finite simple group has the stronger property, known as $\frac {3}{2} …

We say that a finite group G satisfies the independence property if, for every pair of distinct
elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x …

For a finite group G a graph Γ (G) is defined on the elements of G in such a way that two
distinct vertices are connected by an edge if and only if they generate G. Some results and …

For a finite group G let Γ (G) denote the graph defined on the non-identity elements of G in
such a way that two distinct vertices are connected by an edge if and only if they generate G …

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 2-generated soluble group and suppose that $\langle a_1,
b_1\rangle=\langle a_2, b_2\rangle= G $. If either $ G^\prime $ is of odd order or $ G^\prime …

For a finite group G, let Γ (G) denote the graph defined on the non-identity elements of G in
such a way that two distinct vertices are connected by an edge if and only if they generate G …

For a finite group G let Γ (G) denote the graph defined on the non-identity elements of G in
such a way that two distinct vertices are connected by an edge if and only if they generate G …

We investigate the extent to which the exchange relation holds in finite groups $ G $. We
define a new equivalence relation $\equiv _ {\mathrm {m}} $, where two elements are …