On finite groups with prescribed two-generator subgroups and integral Cayley graphs

Abstract

In this paper, we characterize the finite groups $G$ of even order with the property that for any involution $x$ and element $y$ of $G$, $⟨x,y⟩$ is isomorphic to one of the following groups: ${\mathbb{Z}}_2,$ ${\mathbb{Z}}_2^2$, ${\mathbb{Z}}_4$, ${\mathbb{Z}}_6$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_6$ and $A_4$. As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree $3$ have integral spectrum.