LOCALLY NILPOTENT SUBGROUPS OF $\mathrm {GL}_n(D)$

Abstract

Let A be an F-central simple algebra of degree $m^2=\prod _{i=1}^k p_i^{2\alpha _i}$ and G be a subgroup of the unit group of A such that $F[G]=A$ . We prove that if G is a central product of two of its subgroups M and N, then $F[M]\otimes _F F[N]\cong F[G]$ . Also, if G is locally nilpotent, then G is a central product of subgroups $H_i$ , where $[F[H_i]:F]=p_i^{2\alpha _i}$ , $A=F[G]\cong F[H_1]\otimes _F \cdots \otimes _F F[H_k]$ and $H_i/Z(G)$ is the Sylow $p_i$ -subgroup of $G/Z(G)$ for each i with $1\leq i\leq k$ . Additionally, there is an element of order $p_i$ in F for each i with $1\leq i\leq k$ .