How to use group actions

Quick description

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

General discussion

Click here for a brief discussion of group actions. ( 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 $g(hx).$ (This is actually the definition of a left action. Analogously, we can write $xg$ for $\phi(g)(x)$ , in which case we require that $x(gh)=(xg)h$ . This is the definition of a right action.) 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. )

The articles

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.)
To show that a group element is non-trivial, show that it has a non-trivial action or image Quick description ( A group $G$ may be defined so indirectly that it can be difficult to distinguish one group element from another; but often there are ways to map $G$ to a more concrete group, or to make $G$ act on a more concrete space. Using such concrete representations, it becomes easier to distinguish elements in $G$ .)
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.)
Use topology to study your group 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.)