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189-571B: Higher Algebra II

Assignment 1

To be presented: Wednesday, January 17.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Show that there are no (associative) division algebras over R of odd rank.

2. (Matthew Greenberg) Let G be a finite group and let r1,...,rs be representatives for a complete set of isomorphism classes of irreducible complex representations of G, of dimensions n1,...,ns respectively. Show that the C-algebra homomorphism induced by the rj's:
C[G] --> Mn1(C) + ... + Mns(C)
is an isomorphism.

3. (Clotilde Paris de la Bollardiere) Show that
R[D8] = R+R+R+R+M2(R),

R[Q] = R + R+ R+R+ H,

where D8 and Q are the dihedral and quaternion groups of order 8, and H is the ring of Hamilton's quaternions. Conclude that R[Q] is not isomorphic to a direct sum of matrix rings over R.

4. (Martin Caberlin) The augmentation map on the group ring F[G] is the homomorphism which sends
a1 g1+ ... + an gn
a1 + ... + an.
Its kernel is called the augmentation ideal. Show that if F is a field of characteristic p and G is a group of p-power order, then the augmentation ideal I of F[G] is nilpotent, i.e., for all x in I, there exists N such that