G-set:  two equivalent definitions:   G x X --> X with axioms;   
	G --> Bij(X) group homomorphism
G-maps between G-sets

Motivation and defining actions:
	For X object (set, vector space, inner product space, etc.), 
	Aut(X) is a group,  called the group of symmetries of X

	Aut(X) x X --> X natural map is an action
	
	We thus get defining actions of Bij(X) on a set X,
	GL(V) on a vector space V
	O(V) on an inner product space V
	etc.

Representation intuitively means "representing" abstract by the concrete.
	The concrete ones are so called because they motivated the definition
	of the abstract in the first place.

	For groups,  we think of Aut(X) as being concrete. 
	Representation of a group G means gp homomorphism  G --> Aut(X)

	Depending upon the nature of X,  we get various kinds of
	representations:   if X set,  then permutation representation
	if X=V vector space,  then linear representation
	if X=V inner product space,  then orthogonal representation
	etc.

	If G --> Aut(N),  for a group N, then we can form the semi-direct
	product

Explicitly wrote down the two equivalent definitions of linear representation of a group:
	G --> GL(V) group homomorphism
	G x V --> V action map;   g: v-->gv a linear transformation for every g.

Two equivalent definitions of a K-algebra (associative unital K-algebra):
	R ring, also a K-vector space,  underlying additive group of both
	being the same;   R x R --> R multiplicaiton is bilinear

	R ring,  K --> R ring hom with image in centre of R

Algebra homomorphisms:   R --> S ring map that is also K linear;
	R --> S ring map and 
	inclusion of K in R followed by this gives inclusion of K in S

Concrete examples of algebras:  K-End(V):  K-linear endomorphisms
	of a K vector space V

Representations, more commonly called modules, for a K-algebra:
	R --> K-End(V)  K-algebra morphism

X set,  linearization of X:  KX = finite formal linear combinations of X
	Universal property

G group,  KG linearization of G:  becomes a K-algebra
		Given	G --> GL (V) group homomorphism
		have KG --> K-End (V) algebra morphism
		Conversely also.   Thus:
		K-linear representation of G same as KG-module

Approaches to rep theory of G:    character theory
				ring theory
				module theory


problem extras:  semi-direct products of groups  
		Dedekind's linear independence of characters