Groups in action

The raison d'etre of groups is to act, viz the elements of a group behave as automorphisms of a structure. Here, we look at the various kinds of group actions, and the relation between each kind of group action and the underlying group structure.

Definition
The simplest kind of group action is that on a set. A group action of a group $$G$$ on a set $$S$$ is a homomorpihsm $$\rho: G \to Sym(S)$$. In other words, every element of the group induces a permutation on $$S$$ and the product of two elements of the groups induces the composition of the two permutations.

A group action can also be viewed as a map $$G \times S \to S$$ where the image of $$(g,s)$$ is $$\rho(g).s$$. We often write this simply as $$g.s$$. The homomorphism property then translates to:

$$e.s = s$$

and

$$g.(h.s) = (gh).s$$

Before the advent of the abstract group
Historically, in the time of Cayley and Sylvester, groups were not viewed as abstract sets with a multiplication, but rather as groups of transformations on sets. In fact, the term group itself arose from the idea of a group of transformations. For instance, a contemporary of Cayley would define a group as something like: a collection of transformations on a set that is closed under composition and inversion. Today, we look at this as the definition of a faithful group action.

In this historical view, two different faithful group actions of the same group would actually be viewed as different groups. This is roughly analogous to saying that the same actor, in two different plays, is treated as different persons. The identity of the group was thus tied to its action.

After the advent of the abstract group
After the abstract definition of group was laid down by von Dyck and generally accepted by the mathematical community, people started considering the question: given an abstract group, what is the relation bettween its abstract structure, and the various group actions (also called permutation representations) that it can have?

Permutation representations were used, and explored, heavily in the 1880-1900 period, with important results being proved by Sylow (such as the Sylow theorems) and by Burnside and Frobenius.

Definition
A linear representation of a group $$G$$ over a field $$k$$ is a homomorphisms $$\rho: G \to GL(V)$$ where $$V$$ is a $$k$$-vector space, and $$GL(V)$$ is the group of invertible $$k$$-linear maps from $$V$$ to itself.

In other words, a linear representation is an action of a group on a vector space by linear transformations.

The catching-on of linear representations
The study of general linear groups and similar phenomena first gained currency with the work of Sophus Lie, which was mainly on Lie groups over fields like reals or complex numbers. However, they were not initially perceived as a useful tool to study finite groups.