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I have been looking through properties and theorems linked to Abelian groups but so far have had no luck finding anything that would help.

The closest I have is that it may have something to do with "any subgroup of an Abelian group is normal"

My line of argument was if |Aut G|=1 then Inn G = Aut G and so since Inn G is normal then G is normal?

However something tells me this is incorrect.

Thanks in advance for any help