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

TL;DR

This paper characterizes finite even-order groups based on their two-generator subgroups and explores the conditions under which their Cayley graphs of degree 3 have integral spectra, linking group structure to graph spectral properties.

Contribution

It provides a complete characterization of finite groups with specific two-generator subgroup structures and their relation to integral Cayley graph spectra.

Findings

Finite groups with prescribed two-generator subgroups are characterized.

Finite groups with degree 3 Cayley graphs having integral spectra are identified.

The results connect group structure with spectral properties of 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 $G$, $\langle x, y \rangle$ 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.