**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 S_{3}.

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

4. Write down the character table of A_{4} and S_{4}.

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 D_{8} of order eight. What do you notice?

7. Two permutation representations X_{1}
and X_{2} of a finite group G are said to be
*linearly isomorphic, * if L^{2}(X_{1}) and
L^{2}(X_{2}) 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.