TITLE: Structure of antiautomorphisms of a group
QUESTION [5 upvotes]: Surely, the set of automorphisms of a group $G$ becomes another group $\text{Aut}(G)$. Can I say something about the set of $anti$-automorphisms of a group? It is not a group, but I guess there is a theory on it. At least, can I classify the antiautomorphisms in some nontrivial cases, in any sense?

REPLY [11 votes]: Every anti-automorphism is just  $\phi(g)=\rho(g^{-1})$ for some automorphism $\rho$. So the set of all automorphisms and anti-automorphisms are a group, and it is isomorphic to either $\mathrm{Aut}(G)$ (if $G$ is commutative) or  $\mathrm{Aut}(G)\times \mathbb Z/2\mathbb Z$.
Anti-automorphisms are not a group under composition, but as a coset of $\mathrm{Aut}(G)$, you could define a group structure on the set which essentially made it isomorphic to $\mathrm{Aut}(G)$, and not very interesting.
Involution anti-automorphisms just correspond to involution automorphisms.