Finite groups with star-free noncyclic graphs

Abstract

For a finite noncyclic group G, let Cyc(G) be the set of elements a of G such that 〈a, b〉 is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G ∖ Cyc(G), having an edge between two distinct vertices x and y if 〈x, y〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are K1,n-free, where K1,n is a star and 3 ≤ n ≤ 6.

Keywords

• noncyclic graph
• finite group
• star-free
• 05c25
• 20b05