A group representation tries to fit the multiplication in the group into a matrix:
g -> D(g)
is a homomorphism with D(g) a order(G)xorder(G) matrix
g1g2->D(g1)D(g2)=D(g1g2)
Group multiplication -> matrix multiplication
D(e)= Identity.
D(g^)=(D(g))^
if D is an isomorphism => representation is faithful.
=> g1!=g2 => D(g1)!=D(g2)
unfaithful representation => non-trivial kernel, K.
Example:
D2:
D(e)=
| 1 0 |
| 0 1 |
D(a)=
| -1 0 |
| 0 1 |
D(b) =
| 1 0 |
| 0 -1 |
D(ab) =
| -1 0 |
| 0 -1 |
Rotation group of a circle
R(o) =
| cos(o) -sin(o) |
| sin(o) cos(o) |
R(o1)R(o2)=R(o1+o2)