Finite groups with star-free noncyclic graphs

Abstract

For a finite noncyclic group $G$, let $\Cyc(G)$ be a 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 $G\setminus \Cyc(G)$, 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 $K_{1,n}$-free, where $3\le n\le 6$.

10 pages