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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.)