Group theory convention

Question

When I was an undergraduate, in my first courses in group theory the convention was that we always act on the right, so if $f:G\rightarrow H$ is a group homomorphism, then we would write $(g)f$ for the image of $g$ under $f$. Is this usual? I think this may be a convention that comes from finite group theory and discrete mathematics since my department was strong in those fields. I have not seen it in any other field. The book Representations and Characters of Groups by James and Liebeck uses this convention.

Answer 1

There are some books, especially some written in the 60s, that use that convention. Hanna Neumann’s Varieties of Groups has homomorphisms act on the right.

My understanding is that the convention actually comes from Ring Theory: if you like your modules to be left modules, then you want functions to act on the right so that homogeneity looks like associativity: $$(am)f = a(mf).$$ This also makes modules into bimodules, by letting the ring act on the left and the morphisms act on the right.

(It also makes morphisms of left $G$-sets look like that, since you have $(gx)f = g(xf)$ for $x\in X$, $g\in G$ acting on the left, and $f\colon X\to Y$ a $G$-set morphism)

From what I can tell, because “functions acting on the right” has not taken off quite as much, this has actually resulted instead in people working more with right modules (instead of left modules) in more recent times. But this is an impression “one step removed” (I don’t do ring theory, but I have a lot of colleagues who do) so it could be mistaken.