Abstract
Given a finite group G with a normal subgroup N, the simple graph
$\Gamma _{\textit {G}}( \textit {N} )$
is a graph whose vertices are of the form
$|x^G|$
, where
$x\in {N\setminus {Z(G)}}$
and
$x^G$
is the G-conjugacy class of N containing the element x. Two vertices
$|x^G|$
and
$|y^G|$
are adjacent if they are not coprime. We prove that, if
$\Gamma _G(N)$
is a connected incomplete regular graph, then
$N= P \times {A}$
where P is a p-group, for some prime p,
$A\leq {Z(G)}$
and
$\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$
.
Alternating groups as products of four conjugacy classes