Units of a group ring.

Question

Let $\mathbb{Q}$ be the rationals and $G$ a group. Then we consider the group ring $\mathbb{Q}[G]$.

Since the operation on $\mathbb{Q}[G]$ restricted to $G$ is just the group operation, I know that $G$ is a subgroup of the units of $\mathbb{Q}[G]$.

How can we describe all units of $\mathbb{Q}[G]$?

Answer 1

For a general group $G$ this is a hard problem.

Let me mention the partial progress that I remember.

For one thing, if $G$ isn't torsion-free, there is a theorem in Passman's Algebraic structure of group rings that says that the units of $F[G]$ for a field $F$ are trivial iff $G$ is $F_2[C_2]$, $F_2[C_3]$ or $F_3[C_2]$. ($F_i$ denotes a field of order $i$ and $C_i$ the cyclic group order $i$.)

So attention is usually restricted to torsion-free groups, and the open conjecture is the unit conjecture that asks if $F[G]$ has trivial units for any field $F$ and any torsion-free group $G$.

Two good resources I can recommend are Passman's Algebraic structure of group rings and Lam's First course on noncommutative rings I think has some information on this. I don't have copies of Milies' group ring books, but I remember when I consulted them they were very good too.

Finally, through this MO post I found a very interesting slide set for a talk about the difficulty of the conjecture, which looks very good.

Answer 2

This is not answer but some inspiration. In this general setting you've given nothing is known about the group of units of this ring, except for - as you already mentioned - it has a subgroup isomorphic to $G$.

In special cases there are some references I can give you:

Read the Wikipedia article about group rings in general. If $G$ is a finite group, modules over the group ring coincide with linear representations of $G$ and as $\mathbb{Q}[G]$ is a module over itself, it contains every linear representation of $G$. This is called the regular representation.

Furthermore there is an old article of Higman having some nice results in the case of $G$ finite abelian.

Last but not least I'd like this thread in MO which collects some cases over group algebras.

Hope that helps you. Maybe someone else knows something more about units.

Enjoy, Tom