T. Keshavarzipour and M. Mahdavi-Hezavehi: On the non-triviality of $G(D)$ and the existence of maximal subgroups of $GL_1(D) $

keshavarzi_pour@mehr.sharif.edu, mahdavih@sharif.edu

Submission: 2004, Dec 18

Let $D$ be an $F$-central division algebra of index $ n $. Here we investigate a conjecture posed in \cite{haz} that if $ D $ is not a quaternion algebra, then the group $G_0(D)=D^*/{F^*D'}$ is non-trivial. Assume that either $ D $ is cyclic or $ F $ contains a primitive $ p $-th root of unity for some prime $ p|n $. Using Merkurjev-Suslin Theorem, it is essentially shown that if none of the primary components of $ D $ is a quaternion algebra, then $G(D)=D^*/{RN_{D/F}(D^*)D'}\neq 1$. In this direction, we also study a conjecture posed in \cite{ak} or also \cite{m} on the existence of maximal subgroups of $ D^*$. It is shown that if $D$ is not a quaternion algebra with $ i(D)=p^e$, then $D^*$ has a maximal subgroup if either of the following conditions holds: (i) $F$ has characteristic zero, or (ii) $F$ has characteristic $p$, or (iii) $F$ contains a primitive $p$-th root of unity.

2000 Mathematics Subject Classification: 16K20

Keywords and Phrases: division ring, maximal subgroup, splitting field

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