Math 126 Problem Set 1

This problem set is due October 2, 1998.

1. (25 pts.) Let M be an n by n matrix with complex entries and assume that there exists a positive integer N such that M^{N} = 1. Prove that M is diagonalizable. Find a counter-example if we drop the finite order assumption.

2. (25 pts.) Let R be a ring. Let E be a semisimple module over R. Prove that every submodule and every factor module of E is also semisimple.

3. (50 pts.) Let D_{n} denote the dihedral group of order 2n. This is the group generated by two elements g and r which satisfy the relations: g^{n} = 1, r^{2} = 1, and rgr = g^{−1}. Classify the simple representations of D_{n}. (Note that we’ve done the cases n = 1, 2, and 3 in class.)