Im currently taking a course in Group theory and have been told that looking at right actions instead of left actions is a widely adopted convention in Group theory. At least i know that in GAP groups always act from the right. I dont have any problem with looking at right actions, but i am curious as to why this convention has become a thing.
Are there cases where working with right actions is more pleasant? The only example i can think of is the action of $K^n \times GL(n,K) \to K^n; (x,A) \mapsto xA$ where we interpret $K^n$ as row vectors. I guess row vectors are a little more comfortable to type. Are there other reasons?
Mathematical notation is influenced by famous people, more or less. If important people, who write important books, use one type of notation then everyone else will copy it.
The notation $f(x)$ is terrible for most Western authors, who write from left to right. Why on Earth is $f\circ g$ 'do $g$ first then $f$'? Euler notation of $f(x)$ just because popular because big names used it.
In group theory, it became popular among some important people who wrote important books that $(1,2)(1,3)=(1,2,3)$, i.e., $fg$ is 'do $f$ first, then $g$'. This only looks wrong because $f(x)$ is already silly notation when writing left-to-right, and $xf$ is a much more sensible notation, or even $(x)f$ if you have to need the brackets. It also makes conjugation make more sense, because conjugation is $x^g$. Placing the symbol on the right means you have to contort yourself if you don't adopt right notation. Alternatively, conjugation may be written as ${}^g\!x$, which is annoying to type.
There are also group theorists who write right-to-left, and it is not helpful that there are two different notations in the subject.