Free Unit Groups in Algebras

Abstract

Let R be an algebra over a field K, and let G be a finite group of units in R. Suppose that either char K = 0 and G is nonabelian, or K is a nonabsolute field of characteristic π > 0 and G/𝕆_π(G) is nonabelian. Then we show that there are two cyclic subgroups X and Y of G of prime power order, and two special units u _X  ∈ KX ⊆ R and u _Y  ∈ KY ⊆ R, such that ⟨u _X , u _Y ⟩ contains a nonabelian free group. Indeed, we obtain a rather precise description of these units, generalizing an earlier result where R = K[G] was the group algebra of G over K.

Key Words:

• Algebras
• Group algebras
• Units

2000 Mathematics Subject Classification:

• 16S34
• 16U60

Acknowledgments

The first author's research was supported in part by CNPq Grant 302.756/82-5 and Fapesp-Brazil, Proj. Tematico 00/07.291-0. The second author's research was supported in part by NSF Grant DMS-9820271.