Classification actions of finite groups

Question

This is a very elementary question, but I'm having trouble finding anything explicit about it: Consider a finite group $G$ with an action on a finite set $X$. Is there a classification of such actions up to isomorphism (bijections on the set $X$)?

If I had to guess, I would say that we can first decompose $X$ into invariant subsets. The action on each subset is just the action of $G$ onto the cosets of some subgroup $H$. Since I couldn't find anything along this lines, I assume this is wrong; in this case, what would be examples of actions which don't arise in that way?

Answer 1

I would say that we can first decompose $X$ into invariant subsets.

By "invariant" subset do you mean that $G\cdot A\subseteq A$? If so, then you're on the right track. We have to push that idea to the limit, i.e. by taking minimal invariant subsets, or in other words: orbits. Lets add some formalism first.

Given a group $G$ a pair $(X,\cdot)$ is called a $G$-set, if $X$ is a set and $\cdot:G\times X\to X$ is a group action on $X$. With this approach we say that a function $f:X\to Y$ between two $G$-sets is $G$-equivariant if $f(g\cdot x)=g\cdot f(x)$ for any $g\in G$ and $x\in X$. With that two $G$-sets $X$ and $Y$ are $G$-isomorphic if there are $G$-equivariant maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g=id$ and $g\circ f=id$. This can be shown to be equivalent to the fact that there is a $G$-equivariant bijection $X\to Y$.

All of this is done so that we can precisely define what "the same" or "isomorphic" for $G$-sets mean. To catch the true nature of action, regardless of labeling of elements. And with that we have the following:

Lemma. Let $X$ be a $G$-set. Then $X$ is $G$-isomorphic to the disjoint union $\bigsqcup X_i$, where each $X_i$ is $G$-isomorphic to $G/H$ for some subgroup $H\subseteq G$. The decomposition is unique up to $G$-isomorphism. Moreover $G/H$ is $G$-isomorphic to $G/H'$ if and only if $H$ is conjugate to $H'$.

Note that neither $G$ nor $X$ are assumed to be finite. This holds for arbitrary group and $G$-set.