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 GL3(F) of the third degree over the field F of characteristic zero. In contrast to iterative methods, each element of the cyclic subgroup ⟨M⟩ of the group GL3(F) is a linear combination of M0, M, M2, 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 GL3(F) we derive à formula for the cyclic subgroups of prime order p of linear group GL3(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.