We discuss the problem of what the full automorphism 
group of a typical Cayley 
graph or digraph of an abelian group of odd prime-
power order must be.  It 
was shown by Godsil that almost all Cayley digraphs of 
an abelian group G of 
odd prime-power order have automorphism group 
isomorphic to G.  For those 
Cayley digraphs of G of odd prime-power order whose 
automorphism group is 
not G, we show that almost all of them are normal 
Cayley digraphs of G.  That 
is, their automorphism group contains G as a transitive 
normal subgroup.  
Analogous results are also obtained pertaining to the 
automorphism groups of 
Cayley graphs of abelian groups of odd prime-power 
order.

IMFM and Univerza v Ljubljani, 
Fakulteta za matematiko in fiziko
Jadranska 21 (the new building)