1 results
LOCALLY NILPOTENT SUBGROUPS OF
$\mathrm {GL}_n(D)$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 06 November 2024, pp. 1-11
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- Article
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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 
$1\leq i\leq k$.
