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DOI: 10.23671/VNC.2018.2.14722

On Cyclic Subgroups of a Full Linear Group of Third Degree over a Field of Zero Characteristic

Pachev U. M. , Isakova M. M.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.
In this paper, using the concept of the spectrum of a matrix, we give an explicit form for the elements of any cyclic subgroup in the full linear group \(GL_3(F)\) of the third degree over the field \(F\) of characteristic zero. In contrast to iterative methods, each element of the cyclic subgroup \(\langle M \rangle\) of the group \(GL_3(F)\) is a linear combination of \(M^{0}\), \(M\), \(M^{2}\), with coefficients easily computed using determinants of the third order, composed by certain powers of the eigenvalues of the matrix \(M\). In fact, we offer a new approach based on a property of the characteristic roots of the polynomial of the matrix. Note also that we present a method that involves the previously known eigenvalues of the matrix. Finally, basing on the results about the explicit form of the elements of any cyclic subgroup of the group \(GL_3(F)\) we derive à formula for the cyclic subgroups of prime order \(p\) of linear group \(GL_3(K^{(p)})\) over a circular field \(K^{(p)}\) of characteristic zero that is of interest in their own right in the theory of infinite groups.
Keywords: complete linear group, cyclic subgroups, spectrum of a matrix, diagonalizable matrix, \(n\)-circular field, algebraic closure of a field
Language: Russian Download the full text  
For citation: Pachev U. M., Isakova M. M. On cyclic subgroups of a full linear group of third degree over a field of zero characteristic. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 62-68. DOI 10.23671/VNC.2018.2.14722
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