## Represention Theory

Represention theory is a specialisation of group theory that is restricted to vector spaces.

## Classification of Groups

## Conjugacy Classes

There is a 1:1 correspondence between transitive actions and conjugate classes.

Conjugacy classes of subgroups | Transitive actions of G |

collection of subgroups conjugate to some given subgroup. | Transitive = one orbit Any action = sequence of transitive actions. |

## Actions

We can look at the concepts that we used for permutations, now that we are restricting ourselves to linear maps:

### Stabiliser

Subgroup of G consisting of all elements h such that:

h x = x

h is the stabiliser of x

### Orbit

### Centre

## Mat(n,F)

May be any n×n matrix containing **R**, **C**, **H** or
**O** elements.

- Mat(n,
**R**) - Mat(n,
**C**) - Mat(n,
**H**) - Mat(n,
**O**)

The definitions don't make it totally clear whether Mat(2,C) is any 2×2 matrix whose elements are C or is it the matrix algebra over the centre of C

From Wikipedia: "The term center or centre is used in various contexts in

abstract algebra to denote the set of all those elements that commute with

all other elements".

## SL(n, F) The Special Linear Group

Consisting of n×n matrices where each element is of type 'F' with determinant =1.