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We show that almost every Cayley graph Γ of an abelian group G of odd prime-power order has automorphism group as small as possible.  Additionally, we show that almost every Cayley (di)graph Γ of an abelian group G  of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of G (that is, G L  is normal in Aut(Γ)).

Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power order