For finite group $G$ when is $|Aut(G)| < |G|$?

Question

If $G$ is a finite group then we know $|Aut(G)|$ divides $(|G|-1)!$ ; I want to ask , if $G$ is a finite group with more than one element then is it true that $|Aut(G)| < |G|$ ( I know it is always true for cyclic groups) ? If $|Aut(G)| < |G|$ is not always true then can we characterize those groups for which it is true ?

Answer 1

This claim is false in general. You can check yourself that it fails for group of small order (quarterion group, for example). What is true is the order of elements in the automorphism group must be less than the order of the group itself. For a reference see here:

https://mathoverflow.net/questions/1075/order-of-an-automorphism-of-a-finite-group

For proof of the counter-example see here:

Automorphism group of the quaternion group