Why does group theory usually deal with right actions? 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?
 A: One reason is that if you act first by $A$ and then by $B$ on $x$, you might like to write their composition as $AB$, but it is consistent with right action: $xAB$. For left action, you have to write the group elements in the reverse order: $BAx$.
A: 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.
