# Original papers on the subject of group actions

Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. Wikipedia are for current textbooks.

I am interested to find out the original motivation for introducing the concept. That is, whether group action was first introduced in its own right as a binary operation, which is a generalisation of the action of a permutation on a set, or whether the representation of an abstract group by a permutation group was considered first, and then the action defined later as an axiomatisation. (I realise these are equivalent concepts, but I am more interested in precisely which came first - of course this may be a 'chicken and egg' situation, but I just thought I would ask.) Many thanks

Just to clarify, the 'picture' I have in mind when explaining the possible origin of the notion is this: someone looked at the behaviour of a permutation $\sigma$ acting in the natural way on an element $x$ in a set $X$ to give $\sigma(x)$, with its properties such as $\sigma_1(\sigma_2(x))=(\sigma_1\sigma_2)(x)$, and then decided to generalise this to an arbitrary group by inventing a binary operation and forcing the group elements to obey the same sort of relations as the permutations, i.e. for the 'composition of maps' property $g.(h.x) = (gh).x$.

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Would it be surprising if it went the other way around? It seems possible that "group theory" sprung out of the study of permutations. – Thomas Andrews Dec 5 '12 at 14:13

## 2 Answers

I think it all goes back to Abel, Ruffini, Lagrange and of course Galois, although the notion of a group was not totally formalized at this time. What I am 90% sure about, is that the notion of group action (that is transformation groups) came before the notion of abstract group.

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Thanks, I hadn't heard of Ruffini before. I suppose I am thinking more about the more formal definition of a group action as a map $G \times X \rightarrow X$ with the two usual axioms, as opposed to the general idea, which I am sure existed previously in a less formal form. – user50229 Dec 5 '12 at 14:32

I found a link to Arthur Cayley's first paper on group theory with comments, in an answer to a different question on MSE.

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