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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.

Shaun
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Andre of Astora
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  • Never heard about such a thing. I always write $f(g)$, just like I would do with any other function. – Mark May 04 '20 at 22:58
  • I think this is Artin's notation, which is, more exactly, $ g^f$, which is a common for a group action (it's not really a group action here, but if we consider the set of automorphisms of the group $G$, we do have a group action of the set of automorphisms on $G$ – Bernard May 04 '20 at 23:03
  • I remember group actions were indeed written $x^g$ for $g$ acting on $x$. – Andre of Astora May 04 '20 at 23:04
  • Related. – Shaun May 04 '20 at 23:05
  • Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Shaun May 04 '20 at 23:06
  • It could be leas about discipline and more about background. My Polish algebra professor did that. – Randall May 04 '20 at 23:13
  • @VishnuM: It depends: you also have ${}^gx=gxg^{-1}$, which is the “natural” action if you want the map from $G$ to $\mathrm{Aut}(G)$ that sends $g$ to “conjugation by $g$” to be a morphism, since then $hg\mapsto {}^{hg}\Box$ gives a morphism. Sending $g$ to $\Box^g$ gives a right action instead of a left action. – Arturo Magidin May 04 '20 at 23:18

1 Answers1

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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.

Arturo Magidin
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