Course 424
Group Representations I
Dr Timothy Murphy
Arts Block A2039

Friday, 20 January 1989

15.45–17.45

Answer as many questions as you can; all carry the same number of marks.
Unless otherwise stated, all groups are finite, and all representations are finite-dimensional over C.
1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent?
Determine all 2-dimensional representations of S3 up to equivalence, from first principles.
2. What is meant by saying that the representation α is simple?
Determine all simple representations of D4 , from first principles.
3. What is meant by saying that the representation α is semisimple?
Prove that every finite-dimensional representation α of a finite group over C is semisimple.
Show from first principles that the natural representation of Sn in Cn
(by permutation of coordinates) splits into 2 simple parts, for any n >
1.

4. Define the character χα of a representation α.
Define the intertwining number I(α, β) of 2 representations α, β. State and prove a formula expressing I(α, β) in terms of χα , χβ .
Show that the simple parts of a semisimple representation are unique up to order.
5. Prove that every simple representation of an abelian group is 1-dimensional.
Is the converse true, ie if every simple representation of a finite group
G is 1-dimensional, is G necessarily abelian? (Justify your answer.)
6. Draw up the character table of S4 , explaining your reasoning throughout.
Determine also