Math 126, Problem Set 2

This problem set is due October 9, 1998.

1. (25 pts.) Let R be a semisimple ring. Let M be an R-module with M ≠ 0. Using the notation from class, write R = Π_{i=1}^{m}R_{i}, where R_{i} is a simple ring.

Show that

M = ⨁_{i=1}^{m} R_{i}M = ⨁_{i=1}^{m} e_{i}M,

where 1 = ∑_{i=1}^{m} e_{i} and e_{i} ∈ R_{i}.

Show that

R_{i}M = ∑_{N≅L_{i}} N,

where L_{i} is a simple left ideal in R_{i}.

2. (25 pts.) Are groups, up to isomorphism, uniquely determined by their character table? Hint: Compute the character tables for the groups O and D_{4}, where D_{4} is the dihedral group of order 8 and O is the quaternion group. That is, O = {±1, ±i, ±j, ±k} where i^{2} = j^{2} = k^{2} = −1 and ijk = −1.

3. (25 pts.) Compute the character table for S_{4}.

4. (25 pts.) For each n, construct all the simple one and two dimensional representations of the symmetric group S_{n}.