TITLE: Anti-isomorphisms
QUESTION [16 upvotes]: Definition: Let $G,H$ be groups and $f: G \to H$ a function such that $f(gh) =f(h)f(g)$ for any $g,h \in G$. Then we call $f$ an antihomomorphism. (Note the swapped order of $f(h)$ and $f(g)$.)

I was deriving some properties of antihomomorphisms and I found that there were a lot of similarities with the usual homomorphisms:
For example, for any antihomomorphism $f: G \to H$, we have:

$f(e_G) = e_H$
$f(g^{-1}) = f(g)^{-1}$

If we define $ker f$ in the usual way (i.e. $ker f = \{g \in G|f(g) = e_H\}$), we have:

$f$ injective $\iff ker f = \{e_G\}$
$ker f \unlhd G$

If we denote the existence of a bijective antihomomorphism between $2$ groups with $\asymp$, we have:

$G/ker f \asymp Im(f)$

Also interesting:

$G \asymp H$ and $H \asymp F \Rightarrow G \cong F$

I know that the existence of an isomorphism between $2$ groups means that both groups have exactly the same structure. 
So my question is: 

From a group theoretic point of view, what is the use of bijective
  antihomomorphisms (= anti-isomorphisms) between 2 groups. Can we give
  it an interpretation like we have for regular isomorphisms?

REPLY [2 votes]: Sorry if this has already been written in someone's answer, but it's worth pointing out that you don't really need anti-homomorphisms between groups. Given an anti-homomorphism $f : G \rightarrow H$, the function $g \mapsto (f(g))^{-1} = f(g^{-1})$ is automatically an ordinary homomorphism $G \rightarrow H$, and this process is bijective, and sets up a bijection between anti-isomorphisms and ordinary isomorphisms. So basically, as long as we take the "fundamental" anti-isomorphism $$G \rightarrow G$$ $$g \mapsto g^{-1}$$ seriously, we don't need to worry about any of the others (in some sense.)
The situation changes if we're dealing with monoids, in which case anti-homomorphisms become a bit more useful. In this context, they can also be referred to as "contravariant functors" in the sense of category theory, since a monoid is just a category with one object. And, just like how in category theory we can replace each contravariant functor $\mathbf{C} \rightarrow \mathbf{D}$ with a covariant functor $\mathbf{C}^{op} \rightarrow \mathbf{D}$ and vice versa, so too can we do this in the world of monoids. Nonetheless, I personally think that anti-homomorphisms of categories/monoids are pretty useful; it's just easier to think clearly when you're not introducing ops everywhere in order to artificially coerce everything in sight to be covariant.
That's my two cents.