Example
For many groups it is entirely natural to represent the group through matrices. Consider for example the dihedral group D4 of symmetries of a square. This is generated by the two reflection matrices
Here m is a reflection that maps (x,y) to (− x,y), while n maps (x,y) to (y,x). Multiplying these matrices together creates a set of 8 matrices that form the group. As discussed above, we can either think of the representation in terms of the matrices, or in terms of the action on the two-dimensional vector space (x,y).
This representation is faithful - that is, there is a one-to-one correspondence between the matrices and the elements of the group. It is also irreducible, because there is no subspace of (x,y) that is invariant under the action of the group.
Read more about this topic: Representation Theory Of Finite Groups
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