Dieudonné's Determinants and Structure of General Linear Groups Over Division Rings Revisited
Vol. 16 No. 1 (2021) P.21~P.47
DOI: || https://doi.org/10.21915/BIMAS.2021102 |
| ||10.21915/BIMAS.2021102 |
In this partially expository article we revisit the construction of the Dieudonne determinant and structure of general linear groups over division rings. Our main motivation is to understand the underlying theoretic background and the proof of the simplicity of the projective special linear groups $PSL_n(K)$ over a division ring $K$. The latter gives an important family of simple groups of Lie type. The method of proving simplicity here is based on Iwasawa’s argument which proves the simplicity of $PSL_n(F)$, where $F$ is a field. This is simpler than the proof given in E. Artin’s exposition [Geometric Algebra, Interscience Publishers, 1957]. We also fix the relation on the determinants of the transposes of matrices in some literature.
General linear groups, division rings, Dieudonn´e determinants.
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 12E15, 16K20
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Revised : 2021-03-13