Groups, Categories, Representations

A way to think of groups is that they correspond to symmetries. Symmetries of an object satisfy some properties which exactly correspond to the group axioms. An equivalence between an object and its other transformed form is what we mean by symmetry. Act of doing nothing to the object does not transform the object. It corresponds to the identity element of the group and to the reflexive property of equivalence relation. Associative law corresponds to the fact that if you take a symmetry of the object and then again another symmetry of the obtained form we get a new symmetry, and this corresponds to transitivity of equivalence relation. Inverse corresponds to the symmetric property of equivalence relation.

In the above, we viewed a group as a set of transformations on an object which retain its structure. More generally groups occur as the structure-preserving transformation of some objects. These notions of structure and structure-preserving maps can be formalized using Category theory. Category basically is a collection of objects together with a collection of maps between them which we call morphisms. Given an object we can look at the morphism to represent some structure-preserving map to another object. We want an identity morphism between the same object. So we impose some conditions on the morphism between two objects so as to meet our requirements of structure preservation. For instance, we want that if there is a morphism from an object $A$ to another object $B$ , another morphism between $B$ to another object $C$ , we require that there is a morphism between object A and C. So we require that every such pair of morphisms one from $A$ to $B$ another from $B$ to $C$ is mapped to a morphism from $A$ to $C$ . There are similar conditions imposed to make sure we capture the right notion. See http://en.wikipedia.org/wiki/Category_(mathematics)

Once we have these structure-preserving maps we can consider the set of what are called automorphisms of $A$ . These form a group with composition of morphisms being the operation. One can see that most of the groups we occur as structure-preserving map of some object in some category. Now coming to the notion of group action, it is nothing but a homomorphism from the group to Automorphism group of some object in a Category. The homomorphism here is a morphism in the category of Group denoted by Grp. The group action on sets appear here as group actions corresponding to the category of sets. The automorphisms being bijections of sets. So with this group action to Automorphisms of a sets i.e., $Perm(S)$ we associate a permutation of set $S$ to each element of the group. In fact, Cayley’s theorem says that every group can be seen as a subgroup of a $Perm(S)$ for some set $S$ . Historically it was with the study of subgroups of symmetric group, the area of group theory really started. Groups that were at first concretely studied began to get their abstract formalism only very much later when the notions of normal subgroups and quotient groups have started to occupy very crucial role in the study of groups. Although we can just contend with a group as a subgroup of permutation group of some set $X$ , the quotient groups may not be realized as the subgroup of the permutation group of the same set $X$ . So mathematicians started trying to view groups abstractly without referring to any sets. That is, they started viewing them intrinsically. Whenever we want to see how groups manifest as symmetries of other objects we study their actions. This is an extrinsic viewpoint. Although we can just study groups abstractly sometimes we get more information about them by studying their actions on other spaces. For instance, studying action corresponding to automorphism in the category of vector spaces allows us to use tools of linear algebra to understand the structure of the group. Thus we study homomorphism from the group to $GL_n(V)$ . This is the basic idea behind the linear representation theory of groups.