Abstract
Let G be a finite Abelian group and Cay(G,S) the Cayley (di)-graph of G with respect to S, and let A e Aut Cay(G,S) and A_1 the stabilizer of 1 in A. In this paper, we first prove that if A_1 is unfaithful onS then S contains a coset of some nontrivial subgroup of G, and then characterize Cay(G,S) ifA_1^S contains the alternating group on S. Finally, we precisely determine all m-DCIp -groups for 2 ≤ m ≤ p + 1, where p is a prime.
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