Algebra Seminar

Jairo Goncalves, University of Sao Paulo

"Some problems in division rings"

Let $D$ be a division ring with center $k$ and multiplicative group $D^{\bullet}= D \setminus \{0\}$. Moreover, let $N \vartriangleleft D^{\bullet}$ be a noncentral normal subgroup. We address the various cases in which the following conjecture, due to Lichtman, holds:

Conjecture: $N$ contains a free (non cyclic) subgroup.

We also discus a related problem: If $char k \neq 2$, $^{*}$ is a $k$-involution of $D$ and $N^{*}=N$, then $N$ a free subgroup generated by symmetric elements (recall that $x \in D$ is symmetric if $x^{*}=x$).

MC 5403