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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.
Abstract:
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