**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 r_{1},...,r_{s} be
representatives for a complete set of isomorphism classes of irreducible
complex representations of G, of dimensions n_{1},...,n_{s}
respectively.
Show that the C-algebra homomorphism induced by the r_{j}'s:

C[G] --> M_{n1}(C) + ... + M_{ns}(C)

is an isomorphism.

**3**. (Clotilde Paris de la Bollardiere) Show that

R[D_{8}] = R+R+R+R+M_{2}(R),

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

where D_{8} 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

a_{1} g_{1}+ ... + a_{n} g_{n}

to

a_{1} + ... + a_{n}.

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

x^{N}=0.