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189-570A: Higher Algebra I

Assignment 5

Due: Friday, November 26.

1. Lang, Chapter XVIII, Exercise 1, parts (a), (b), and (c).

2. Write down the character table of S3.

3. Lang, Chapter XVIII, Exercise 2, parts (a) to (g).

4. Write down the character table of A4 and S4.

5. Lang, Chapter XVIII, Exercise 3, parts (a) to (b).

6. Write down the character tables for the quaternion group Q and the dihedral group D8 of order eight. What do you notice?

7. Two permutation representations X1 and X2 of a finite group G are said to be linearly isomorphic, if L2(X1) and L2(X2) are isomorphic complex representations of G. Find an example of a group G, and two permutation representations of G, which are linearly isomorphic but not isomorphic as G-sets.