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I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures.

As far as I can tell in the 19th century group actions were much more implicit than they are today and were strongly tied to the group itself, as a subgroup of a permutation group. But the way group actions are presented today (at the very least as they were presented to me) is much more axiomatic and considers group actions and groups as separate algebraic structures.

When and how did this change (if it changed at all) and how did group actions become so important?

I'm also looking for books or articles on this topic, so it would be very much appreciated if anyone has a recommendation, as the books on the history of group theory I have skimmed don't mention the development of group actions.

Mauricio
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paidresolution
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    This question was posted on Math Stack Exchange, where it was suggested to be posted on HSM – J. W. Tanner Jan 31 '23 at 02:40
  • Since historicaly (abstract) group's elements are permutations in disguise, then "obviously" $e\cdot x=x$ and $g\cdot(h\cdot x)=(gh)\cdot x$, precisely like $Id(x)=x$ and $f(g(x))=(fg)(x)$. Therefore, I expect that group actions were introduced "the day after" Cayley(?) defined in the 50's of XIX century the notion of abstract group by mimicking the properties of the set of all the bijections on a set. – citadel Jan 31 '23 at 20:51
  • For example, did Syolw explicitly used group actions in his original proofs? This would date at least back to the 70's of the XIX century their birth. – citadel Jan 31 '23 at 21:01
  • @Devo I looked at Burnsides "Theory of Groups of Finite Order" and he discusses groups "acting" on things only when the group is a subgroup of a symmetric group (which he calls a substitution group). He even has the modern notion of transitivity but group actions as their own structure or object are not mentioned. – paidresolution Jan 31 '23 at 22:44
  • And the original proof of the Sylow Theorems is in French, but The Early Proofs of Sylow's Theorem claims that he only considered subgroups of a permutation group. – paidresolution Jan 31 '23 at 22:48
  • Maybe a good "marker" to catch the "transition point" you are asking about, could be the first explicit mention of the Orbit-Stabilizer theorem? – citadel Feb 01 '23 at 06:10
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    Second (1911) edition of Burnside's book has chapter XII titled On Representation of Group of Finite Order as a Permutation Group, where he gives the definition of representation (borrowed from Frobenius), and proceeds to define the stabilizer subgroup. Isn't this pretty much group action except for the word "action"? The language of "structures" and "actions" spread from Bourbaki's Éléments (1939ff). Algebra I had a section on group actions in particular. – Conifold Feb 02 '23 at 04:04
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    My impression is that the notion "permutation representation" has been around for some time before "group action" took over. A "permutation representation" is a homomorphism of $G$ to a permutation group on some set $X$, which is really the same a an action of $G$ on $X$. It would also be interesting to know where the term "group action" was used the first time. – azimut Feb 03 '23 at 18:20

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