Finite groups with quadratic splitting fields for all Cayley graphs

Abstract

For a graph $\hat{I}“$, the splitting field of $\hat{I}“$ is defined as the splitting field of the characteristic polynomial of $\hat{I}“$ over rationals. The algebraic degree of $\hat{I}“$ is defined by the extension degree of its splitting field over rationals. Let $k$ be a positive integer. We call a finite group $G$ \textit{Cayley $k$-integral} if, for every inverse-closed subset $S$ of $G$, the algebraic degree of the Cayley graph $\Cay(G,S)$ does not exceed $k$. We give a complete classification of all finite Cayley $2$-integral groups. It is shown that a finite abelian group is Cayley $2$-integral if and only if it is isomorphic to one of the following forms: $G\cong {\mathbb{Z}}_2^r\times {\mathbb{Z}}_5^s$, ${\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s\times {\mathbb{Z}}_8^t$, or ${\mathbb{Z}}_2^r\times {\mathbb{Z}}_3^s\times {\mathbb{Z}}_{12}^t$, where $r,s,t\ge 0$. Furthermore, we prove that the set of finite non-abelian Cayley $2$-integral groups consists of the infinite family $Q_8\times {\mathbb{Z}}_2^n$, with $n\ge 0$, and $22$ specific groups.

22 pages