WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
Extended Special Linear group $$ESL_{2}(\mathbb{F})$$ and matrix equations in $$SL_{2}(\mathbb{F})$$, $$SL_{2}(\mathbb{Z})$$ and $$GL_{2}(\mathbb{F}_{p})$$
Authors: ,
Abstract: The problem of roots existence for different classes of matrix such as simple and semisimple matrices from $$SL_{2}(\mathbb{F})$$, $$SL_{2}(\mathbb{Z})$$ and $$GL_{2}(\mathbb{F})$$ are solved. For this purpose, we first introduced the concept of an extended special linear group $$ESL_{2}(\mathbb{F})$$, which is generalisation of the matrix group $$SL_{2}(\mathbb{F})$$, where $$\mathbb{F}$$ is arbitrary perfect field. The group of unimodular matrices and extended symplectic group $$ES_{p2}(\mathbb{R})$$ are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in $$GL_{2}(\mathbb{F})$$, where $$\mathbb{F}$$ is an arbitrary field. The method of matrix factorisation is outlined. We show that $$ESL_{2}(\mathbb{F})$$ is a set of all square matrix roots from $$SL_{2}(\mathbb{F})$$ except of that established in our root existence criterion.
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Keywords: extended special linear group, extended symplectic group, splittable extension, set of squares in matrix group, criterion of square root existing in $$SL_{2}(\mathbb{F}_{p})$$, relations and group generators, matrix factorization
Pages: 643-659
DOI: 10.37394/23206.2024.23.68