The characters of a representation D of G is:
X(g)=Tr(D(g))=trace of the D(g)'s

Theorem:
* a,b in G with a~b => X(a)=X(b) (their characters are the same)

Proof:
D(b)=D(g^ag)=D(g^)D(a)D(g)
X(b)=Tr(D(b))=Tr(D(g^)D(a)D(g))
=> Tr(D(b))=Tr(D(a))

* two equivalent representations of G have the same character

Proof as above

* The character of reducible representations is the sum ov the characters of the component representations

X(g) = X1(g)+X2(g)

* Character of e = dimension of group.

* a finite group G has k inequivalent irreducible representations D1, D2,..,Dk
and l conjugacy classes C1,C2,...,Cn with k=l.