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

Part of: Other groups of matrices Basic linear algebra Division rings and semisimple Artin rings

Published online by Cambridge University Press:  06 November 2024

R. FALLAH-MOGHADDAM Open the ORCID record for R. FALLAH-MOGHADDAM [Opens in a new window]  and
H. R. DORBIDI Open the ORCID record for H. R. DORBIDI [Opens in a new window]
R. FALLAH-MOGHADDAM*
Affiliation:
Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran
H. R. DORBIDI
Affiliation:
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft 78671-61167, Iran e-mail: hr_dorbidi@ujiroft.ac.ir
*
e-mail: r.fallahmoghaddam@cfu.ac.ir
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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 $1\leq i\leq k$.

Keywords

division ringlocally nilpotent subgroupcentral product of groupscrossed product

MSC classification

Primary: 16K20: Finite-dimensional
Secondary: 15A30: Algebraic systems of matrices 20H25: Other matrix groups over rings

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 112 , Issue 1 , August 2025 , pp. 128 - 138
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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