# Writing functions on the right

In group theory, writing functions on the right is a common, though not universal practice.

Thus, given mappings $$f$$, $$g$$ and group element $$\alpha$$, one might write $$\alpha f$$ and $$\alpha (f \circ g)$$ for $$f(\alpha)$$ and $$f(g(\alpha))$$ respectively.

I can see how this makes sense in group theory, especially in relation to permutations.

I am unable to locate any information on when this practice was introduced and I suspect that it may be rather difficult to identify its origins.

Q: When was the practice of writing functions on the right introduced in group theory?

• Related: History of $f \circ g$ – Dave L Renfro Nov 27 '19 at 9:12
• The instance of this I know: if $\sigma$ is an automorphism of a field, then write $x^{\sigma}$ for the image of element $x$ under $\sigma$. So composition is opposite to the usual: $x^{\sigma \tau} = (x^\sigma)^\tau$. – Gerald Edgar Nov 27 '19 at 19:12
• @GeraldEdgar Hi Gerald, Yes, I've seen that notation myself. A previous comment, now deleted, cited a 1929 paper by Noether where she writes group operators on the right. I suspect that this may be the source of the practice of writing mappings on the right in group theory. – Nick Nov 27 '19 at 19:50
• @DaveLRenfro Thanks for your reference and congratulations on having such a sound memory - recalling a posting from 2012. – Nick Nov 27 '19 at 19:51
• As far as memory, that was google. I knew that on several occasions I've mentioned Herstein used an order backwards than standard in his Topics in Algebra book, so I googled my name along with Herstein and composition, then I picked the most significant of the results (in terms of answers and discussion). It was only after I'd written my comment that I realized you were talking about where to place the $x$ (I hadn't realized this in my quick glance at your question before googling), so I then edited "See" (or whatever I began with) to "Related". – Dave L Renfro Nov 28 '19 at 8:46