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Paper IPM / M / 16716  


Abstract:  
Given a finite group G with a normal subgroup N, the simple graph Γ_{G}( N ) is a graph whose vertices are of the form x^{G}, where x ∈ N\Z(G), and x^{G} is the Gconjugacy class of N containing the element x. Two vertices x^{G} and y^{G} are adjacent if they are not coprime. In this article we prove that, if Γ_{G}(N) is a connected incomplete regular graph, then N = P ×A where P is a pgroup, for some prime p, A ≤ Z(G), and Z(N)\not = N∩Z(G).
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