This article provides a list of different ways to study groups by their actions.
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 can be thought of as the group of rotations of a regular -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 is a group, and is a set, then an action of on is a homomorphism from to the set of all bijections from to That is, for each is a bijection from to and these bijections need to compose in a way that reflects the multiplication in : we need to be for every and every If we are dealing with just one action, it can be nicer to write instead of That is, we think of as actually equalling a bijection rather than being transformed into one. Then the rule is that should equal (This is actually the definition of a left action. Analogously, we can write for , in which case we require that . 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 have the same effect on
Sometimes the set is just a finite set, in which case a bijection from to is naturally thought of as a permutation, and the action of as a homomorphism from to the symmetric group on But often 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 to 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.)
To show that a group element is non-trivial, show that it has a non-trivial action or image Quick description ( A group may be defined so indirectly that it can be difficult to distinguish one group element from another; but often there are ways to map to a more concrete group, or to make act on a more concrete space. Using such concrete representations, it becomes easier to distinguish elements in .)
Representation theory Quick description ( When the set 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 , then it can help to realize as the fundamental group of a topological space. This works best when is infinite and discrete, and especially if is finitely presented and torsion-free.)