*-Group identities on units of division rings

Abstract

Let D be a division ring with infinite center F. By a well known result of Amitsur, if $\mathcal{U}(D)$ satisfies a group identity, then D is commutative. Now assume that D has an involution * of the first kind. In this paper, among other results, we show that if $\mathcal{U}(D)$ satisfies a *-group identity, then either D is commutative or dim_F D = 4 and * is of the symplectic type. As a result, let N be a *-invariant normal subgroup of $\mathcal{U}(D)$ such that all symmetric elements of N are central (this is the case when, for example, each symmetric element of N is bounded Engel). Then either N is central or ${\dim }_{F}D=4$ and * is of the symplectic type.

Keywords:

• Division ring
• Engel group
• group identity
• involution

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16K40
• 16R50
• 16U60
• 16W10

Additional information

Funding

The first author is funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number B2020-18-02.