Group action
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Needs fleshing out and neatening up, I'd like to introduce right group actions in a different way to left, however in my current attempt they're the same length!
Contents
Defintion
A (left) group action of a group [ilmath](G,*)[/ilmath] on a set [ilmath]X[/ilmath] is a mapping^{[1]}:
 [ilmath](\cdot):G\times X\rightarrow X[/ilmath]^{[Note 1]} defined by [ilmath](\cdot):(g,x)\mapsto g\cdot x[/ilmath] such that:
 [ilmath]\forall x\in X[1\cdot x=x][/ilmath] (where [ilmath]1[/ilmath] is the identity element of [ilmath](G,*)[/ilmath] group) and
 [ilmath]\forall g,h\in G\ \forall x\in X[g\cdot(h\cdot x)=(g*h)\cdot x][/ilmath]
Notations for [ilmath]g\cdot x[/ilmath] include [ilmath]gx[/ilmath] and [ilmath]{}^gx[/ilmath]
A right group action^{[1]} is almost exactly the same, just the other way around; defined by [ilmath](\cdot):X\times G\rightarrow X[/ilmath] given by [ilmath](\cdot):(x,g)\mapsto x\cdot g[/ilmath] which must satisfy [ilmath]\forall x\in X[x\cdot 1=x][/ilmath] and [ilmath]\forall g,h\in G\ \forall x\in X[(x\cdot g)\cdot h=x\cdot(g*h)][/ilmath].
Notations for [ilmath]x\cdot g[/ilmath] include [ilmath]xg[/ilmath] and [ilmath]x^g[/ilmath]
See also
Notes
 ↑ I have written [ilmath](\cdot):G\times X\rightarrow X[/ilmath] rather than the usual [ilmath]\cdot:G\times X\rightarrow X[/ilmath] notation for functions to make it clearer that there is a dot there; this notation isn't new or different, it's just because a lone [ilmath]\cdot[/ilmath] looks out of place.
References
 ↑ ^{1.0} ^{1.1} Abstract Algebra  Pierre Antoine Grillet

