2
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ Related: History of $f \circ g$ $\endgroup$ – Dave L Renfro Nov 27 '19 at 9:12
  • $\begingroup$ 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: $x^{\sigma \tau} = (x^\sigma)^\tau$. $\endgroup$ – Gerald Edgar Nov 27 '19 at 19:12
  • $\begingroup$ @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. $\endgroup$ – Nick Nov 27 '19 at 19:50
  • $\begingroup$ @DaveLRenfro Thanks for your reference and congratulations on having such a sound memory - recalling a posting from 2012. $\endgroup$ – Nick Nov 27 '19 at 19:51
  • $\begingroup$ 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". $\endgroup$ – Dave L Renfro Nov 28 '19 at 8:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.