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 ) 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. )