Spectra of Cayley graphs

Let $G$ be a group and $Ssubseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}={s^{-1}mid sin S}$. Then {it the Cayley graph ${rm Cay}(G,S)$} is an undirected graph $Gamma$ with the vertex set $V(Gamma)=G$ and the edge set $E(Gamma)={(g,gs)mid gin G, sin S}$. A graph $Gamma$ is said to be {it integral} if every eigenvalue of the adjacency matrix of $Gamma$ is integer. In the paper, we prove the following theorem: {it if a subset $S=S^{-1}$ of $G$ is normal and $sin SRightarrow s^kin S$ for every $kin mathbb{Z}$ such that $(k,|s|)=1$, then ${rm Cay}(G,S)$ is integral.} In particular, {it if $Ssubseteq G$ is a normal set of involutions, then ${rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {it if $G=A_n$ and $S={(12i)^{pm1}mid i=3,dots,n}$, then ${rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in Kourovka Notebook.