On cyclic subgroups of a full linear group of third degree over a field of zero characteristic

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 а 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.