MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g.

I wonder whether more is known about unitary units: Given a group homomorphism $w\colon G \rightarrow \mathbb{Z}/2\mathbb{Z}=\left\{\pm 1\right\}$ we can define an involution on $K[G]$ via $g\in G \mapsto \overline{g}=w(g)g^{-1}$ . The unitary units are those elements $x$ of the group ring with $\overline{x}x=1$. Do we know they are trivial for torsion-free groups?

share|cite|improve this question
In your definition it would be better to replace $\mathbb{Z}/2\mathbb{Z}$ by the subgroup $\{1,−1\}$ of $(K^\times, .)$. These involutions were introduced by Novikov. To the best of my acknowledge, there are no results concerning a unitary version of the unit conjecture. – Salvatore Siciliano Nov 1 '11 at 14:24
@Joerg: Thank for for posting the link to the very interesting slides! If you haven't already seen it, you may find my related post from yesterday interesting:… – Johan Öinert Nov 1 '11 at 15:15
Let's make it more specific: do we have any idea about the unitary unit or unit conjecture for the finite fields $K=\mathbb{Z}/p\mathbb{Z}$? – Joerg Sixt Nov 2 '11 at 9:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.