# A generalization of the anti-automorphisms of a group

Recently, there was a question about anti-automorphisms of a group $G$. In this case, the inversion map $\iota : G \rightarrow G : x \mapsto x^{-1}$ is an anti-automorphism and $\iota \mathrm{Aut}(G)$ is the coset of all anti-automorphisms.

Let $K$ be a permutation group of degree $n$ and let $\pi \in K$. A natural generalization is a $\pi$-automorphism of the group $G$, where we say that a bijection $\phi : G \rightarrow G$ is a $\pi$-automorphism if for all $x_1, \ldots, x_n \in G$,

$$\phi(x_1 \cdots x_n) = \phi(x_{\pi(1)}) \cdots \phi(x_{\pi(n)})$$

We'll say that $\phi$ is a $K$-automorphism if it is a $\pi$-automorphism for some $\pi \in K$.

Now it's clear that the $K$-automorphisms of $G$ form a group and it's not hard to tell what the cosets of $\mathrm{Aut}(G)$ are in this group.

What I'm wondering if there are examples of groups that have interesting $K$-automorphisms or if the structure of the $K$-automorphisms is understood.

• You can certainly write down this definition, but you should motivate it by giving some explicit non-trivial examples and then ask a specific question. But it seems that you want others to do this for you. That's OK, but please remember next time that such considerations should be made in advance of calling something "a natural generalization". – Martin Brandenburg Apr 6 '15 at 21:50
• @MartinBrandenburg I asked this question because no non-trivial examples besides anti-automorphisms came to mind. – Qudit Apr 6 '15 at 22:01
• Yes, I get that, but why do you want to have examples? Is there any purpose of defining "$K$-automorphisms"? Don't get me wrong, it looks interesting, but I don't know if it really is. Any piece of motivation would make your question better and attract more readers. – Martin Brandenburg Apr 6 '15 at 22:08
• @MartinBrandenburg It's just something I got to wondering about after reading the anti-automorphism question. It's a natural question in my humble opinion. – Qudit Apr 6 '15 at 22:13
• It is probably more natural to consider $K$-homomorphisms between arbitrary groups. This way we get a category whose objects are groups and whose morphisms are $K$-homomorphisms. This is because if $f : G \to G'$ is a $\pi$-homomorphism and $g : G' \to G''$ is a $\sigma$-homomorphism, then $g \circ f : G \to G''$ is a $\pi\circ \sigma$-homomorphism (not $\sigma \circ \pi$, I think). – Martin Brandenburg Apr 6 '15 at 22:25

An example of a $\langle (1\,2\,3)\rangle$-automorphism is $f(x)=ex$, where $e$ is some central involution.
• Thanks for this example (and +1). It is more interesting than the ones I had thought of, but what I'm wondering is if there is a $K$-automorphism that somehow exploits the group structure to reorder things beyond just using that some element is in the center. – Qudit Apr 6 '15 at 23:42