The Cayley graphs ${\mathcal{G}}_m(G)=Cay(G^m,S)$ of Cartesian powers of the finite group G with respect to canonical symmetric subsets $S=S(m)\subset G^m$ are investigated. It is proved that for each $m>1$ the graph ${\mathcal{G}}_m(G)$ determines the group G up to isomorphism. The groups of automorphisms $\mathbf{Aut}({\mathcal{G}}_m(G))$ are determined. It is shown that if G is a non-abelian group, then $\mathbf{Aut}({\mathcal{G}}_m(G))\simeq (G^m⋊\mathbf{Aut}(G))⋊D_{m+1}$, where $D_{m+1}$ is the dihedral group of order $2m+2$. If G is an abelian group (with some exceptions for $m=3$), then $\mathbf{Aut}({\mathcal{G}}_m(G))\simeq G^m⋊(\mathbf{Aut}(G)\times S_{m+1})$, where $S_{m+1}$ is the symmetric group of degree $m+1$. Relations between Cayley graphs ${\mathcal{G}}_m(G)$ and Bergman-Isaacs Theorem on rings with fixed-point-free group actions are discussed.

凯莱图 ${\mathcal{G}}_{米}(G)=C\mathrm{一种}是(G^{米},秒)$有限群G关于正则对称子集的笛卡尔幂$秒=秒(米)\subset G^{米}$被调查。证明对于每个$米>1$ 图 ${\mathcal{G}}_{米}(G)$确定群G直至同构。自同构群$\mathbf{自动}({\mathcal{G}}_{米}(G))$被确定。证明如果G是一个非阿贝尔群，则$\mathbf{自动}({\mathcal{G}}_{米}(G))\simeq (G^{米}⋊\mathbf{自动}(G))⋊D_{米+1}$， 在哪里 $D_{米+1}$ 是有序的二面体群 $2米+2$. 如果G是一个阿贝尔群（对于$米=3$）， 然后 $\mathbf{自动}({\mathcal{G}}_{米}(G))\simeq G^{米}⋊(\mathbf{自动}(G)\times {秒}_{米+1})$， 在哪里 ${秒}_{米+1}$ 是度的对称群 $米+1$. Cayley 图之间的关系${\mathcal{G}}_{米}(G)$ 讨论了具有不动点自由群作用的环上的 Bergman-Isaacs 定理。