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If $G$ is a trivial group , obviously $Aut(G)$ is trivial . Does the converse hold $?$ .
When we are given that a group $G$ has trivial automorphism group , can we conclude that the group will be trivial $?$ .
I guess not . For one thing , we know that $$G/Z(G)\cong Inn(G)$$. So , $Aut(G)$ being trivial would imply that $$G\cong Z(G)$$ i.e. $G$ is commutative . But I see nothing about $G'$s being trivial .
The problem is that while I cannot prove it , I could not find a counter example either , from among the groups I am familiar with .
What is the non-trivial commutative group that has trivial automorphism group $?$