Up until now I thought a group action of a group $G$ on a set $X$ was a homomorphism fron the group to the symmetric group on $X$. But today I took a class on geometric group theory and several things seem to imply that the professor thinks it‘s rather an antihomomorphism. ( Beware : I think of the geoup operation in the symmetric group of „$fg$ is first $g$ then $f$“)
Do the conventions differ regarding which research area you are in?
Example (everything quoted from the lecture) Let $H\leq G$ be a subgroup of $G$ and let $X$ be the set of right cosets of $H$. Then $\rho\colon G \to Sym(X), g \mapsto \rho(g)$, where $\rho(g(Hx))=Hxg$ is a transitive right-action. BEWARE : $\lambda’\colon G\to Sym(Y)$, where $Y$ is the set of left cosets of $H$ and $\lambda’(g)(xH)=gxH$ is not a homomorphism and so no group action.
As I understand it it is just the other way around.
When it comes to group actions, the conventions seem to depend more on geography or personal preference than on applications, although this is anecdotal.
Personally, I always think of a group as acting on the left, since each element of the group represents a function $X \to X$, and typically we use the notation $g(x)$ for the function $g$ acting on the element $x$. Since I'm used to composing functions right to left, I read $fg$ as "$g$ then $f$", so that $fg(x)$ is exactly $f(g(x))$. It follows that the convention I assume when multiplying in the symmetric group is the reverse of yours, and that group actions are homomorphisms into the symmetric group for me.
My Italian professor apparently uses the same convention you do for the symmetric group, and thinks of symmetric group elements as acting on the right, but still thinks of other functions as acting and composing on the left, which lead to some confusion about the order that operations should appear when group actions were interwoven with functions.
Ultimately any convention is fine as long as you stick to it. Whether a group action corresponds to a homomorphism or an antihomomorphism depends entirely on the conventions chosen and on whether you choose the axiom $fg(x)=f(g(x))$ or $fg(x)=g(f(x))$ for your group actions.
Finally, don't forget that every left group action is also a right group action by inverting: We can define $(x)f = f^{-1}(x)$ to get $(x)fg = (fg)^{-1}(x) = g^{-1}f^{-1}(x) = ((x)f)g$. This is the reason the choice of convention isn't especially important in practice.