Revision of How to use group actions from Mon, 20/04/2009 - 08:00

Quick description

This article provides a list of different ways to study groups by their actions.

General discussion

Many groups arise naturally as groups of transformations of one kind or another. For example, the cyclic group of order $m$ can be thought of as the group of rotations of a regular $m$ -gon. But even a group that arises in a different way can often be thought of very fruitfully as a group of transformations: it's just that one has to find a mathematical object to be transformed.

More formally, if $G$ is a group, and $X$ is a set, then an action of $G$ on $X$ is a homomorphism $\phi$ from $G$ to the set $S(X)$ of all bijections from $X$ to $X.$ That is, for each $g,$ $\phi(g)$ is a bijection from $X$ to $X,$ and these bijections need to compose in a way that reflects the multiplication in $g$ : we need $\phi(gh)(x)$ to be $\phi(g)(\phi(h)(x))$ for every $g,h\in G$ and every $x\in X.$ (If we are dealing with just one action, it can be nicer to write $gx$ instead of $\phi(g)(x).$ That is, we think of $g$ as actually equalling a bijection rather than being transformed into one. Then the rule is that $(gh)x$ should equal $g(hx).$ ) Note that using the word "homomorphism" rather than "isomorphism" is intentional here: perhaps surprisingly, actions are often very useful even if different elements of $G$ have the same effect on $X.$

Sometimes the set $X$ is just a finite set, in which case a bijection from $X$ to $X$ is naturally thought of as a permutation, and the action of $X$ as a homomorphism from $G$ to the symmetric group on $X.$ But often $X$ has more structure: it might be a vector space, or a topological space, say. Then the interesting actions are the ones where the bijections from $X$ to $X$ are the structure preserving ones: invertible linear maps in the case of vector spaces, and homeomorphisms in the case of topological spaces.

Proving results by letting a group act on a finite set Quick description ( This article discusses various ways of deducing facts about groups by choosing appropriate actions on finite sets. Sometimes the group itself acts, and sometimes another group acts on a set that is defined in terms of the first group.)
Representation theory Quick description ( When the set $X$ on which a group acts is a vector space and the bijections are linear, we have what is called a representation of the group. Representations are a very powerful way of studying groups. They also have many other applications, and representation theory is regarded as a branch of mathematics in its own right.)
Actions on topological spaces Quick description ( If you want to study a group $G$ , then it can help to realize $G$ as the fundamental group of a topological space. This works best when $G$ is infinite and discrete, and especially if $G$ is finitely presented and torsion-free.)