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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
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2 Answers 2

up vote 4 down vote accepted

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
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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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