Geodesic
On cyclic subgroups of a full linear group of third degree over a field of zero characteristic
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 62-68 
U. M. Pachev; M. M. Isakova. On cyclic subgroups of a full linear group of third degree over a field of zero characteristic. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 62-68. http://geodesic.mathdoc.fr/item/VMJ_2018_20_2_a7/
@article{VMJ_2018_20_2_a7,
     author = {U. M. Pachev and M. M. Isakova},
     title = {On cyclic subgroups of a full linear group of third degree over a field of zero characteristic},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {62--68},
     year = {2018},
     volume = {20},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2018_20_2_a7/}
}
TY  - JOUR
AU  - U. M. Pachev
AU  - M. M. Isakova
TI  - On cyclic subgroups of a full linear group of third degree over a field of zero characteristic
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2018
SP  - 62
EP  - 68
VL  - 20
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2018_20_2_a7/
LA  - ru
ID  - VMJ_2018_20_2_a7
ER  - 
%0 Journal Article
%A U. M. Pachev
%A M. M. Isakova
%T On cyclic subgroups of a full linear group of third degree over a field of zero characteristic
%J Vladikavkazskij matematičeskij žurnal
%D 2018
%P 62-68
%V 20
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2018_20_2_a7/
%G ru
%F VMJ_2018_20_2_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Pachev U. M., Shokuev V. N., “Cyclic subgroups of the group $GL_2 (F)$”, Izvestia KBSC of RAS, 7:2 (2001), 72–74 (in Russian)

[2] Shokuev V. N., “Cyclic subgroups of the group $GL_3(F)$”, Izvestia KBSC of RAS, 7:2 (2001), 75–77 (in Russian)

[3] Zhemuhova M. Z., Pachev U. M., “Cyclic subgroups of second degree full linear group over a field of the zero characteristic”, Vladikavkaz Math. J., 13:3 (2011), 17–21 (in Russian) | MR

[4] Voevodin V. V., Kuznecov Ju. A., Matrices and Calculations, Nauka, M., 1984 (in Russian) | MR

[5] Lancaster P., Theory of Matrices, Nauka, M., 1973 (in Russian) | MR

[6] Pachev U. M., “Cyclic Subgroups of the Group $GL_3 (F )$ over a Field of Characteristic Zero”, Proceedings of the International Scientific Conference «Actual Problems of Applied Mathematics and Physics», Izd-vo IPMA KBNC RAN, Nalchik, 2017, 167–168 (in Russian) | MR

[7] Faddeev D. K., Faddeeva V. N., Computational Methods of Linear Algebra, Fizmatgiz, M., 1963 (in Russian) | MR

[8] Lidl R., Niederreiter H., Finite Fields, v. 1, Mir, M., 1988 (in Russian) | MR

[9] Kostrikin I., Introduction to Algebra, Nauka, M., 1977 (in Russian) | MR