Geodesic
On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic
Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 399-407.

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

The definition of saturation condition was formulated at the end of the last century. Saturation condition has become useful in study of infinite groups. A description of various classes of infinite groups with various variants of saturating sets was obtained. In particular, it was found that periodic groups with a saturating set consisting of finite simple non-Abelian groups of Lie type, under the condition that ranks of groups in saturation set are bounded in the aggregate, are precisely locally finite groups of Lie type over a suitable locally finite field. A natural step in further research was the rejection of the periodicity condition for the group under study, and the rejection of the structure of the saturating set as a set consisting of finite simple non-Abelian groups of Lie type with ranks bounded in the aggregate. In this paper, we consider mixed Shunkov groups (i.e., groups that contain both elements of finite order and elements of infinite order). It is well known that the Shunkov group does not have to have a periodic part (i.e., the set of elements of finite order in the Shunkov group is not necessarily a group). As a saturating set, we consider the set of full linear groups of degree 2 over finite fields of even characteristic. The lack of analogues of known results V. D. Mazurova on periodic groups with Abelian centralizers of involutions for a long time did not allow us to establish the structures of the Shunkov group with the saturation set mentioned above. In this paper, this difficulty was overcome. It is proved that a Shunkov group saturated with full linear groups of degree 2 is locally finite and isomorphic to a full linear group of degree 2 over a suitable locally finite field of characteristic 2.
Keywords: Shunkov group, groups saturated with given set of groups. 
@article{CHEB_2019_20_4_a25,
     author = {A. A. Shlepkin},
     title = {On the periodic part of the {Shunkov} group saturated with linear groups of degree 2 over finite fields of even characteristic},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {399--407},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a25/}
}
TY  - JOUR
AU  - A. A. Shlepkin
TI  - On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic
JO  - Čebyševskij sbornik
PY  - 2019
SP  - 399
EP  - 407
VL  - 20
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a25/
LA  - en
ID  - CHEB_2019_20_4_a25
ER  - 
%0 Journal Article
%A A. A. Shlepkin
%T On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic
%J Čebyševskij sbornik
%D 2019
%P 399-407
%V 20
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a25/
%G en
%F CHEB_2019_20_4_a25
A. A. Shlepkin. On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic. Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 399-407. http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a25/

[1] Wei C., Go B., Lytkina D. V., Mazurov V. D., “A characterization of locally finite simple groups of type 3D4 over fields of odd characteristics in the class of periodic groups”, Siberian Math. J., 59:5 (2018), 1013–1019 | MR

[2] Joo C., Lytkina D.\.,V., Mazurov V. D., “A characterization of locally finite simple groups of type G2 over fields of odd characteristics in the class of periodic groups”, Mat. notes, 105:4 (2019), 519–525 | MR

[3] Kargapolov M. I., Merzlyakov Yu. I., Fundamentals of Group Theory, Sciene, M., 1982, 288 pp. | MR

[4] Shlepkin A. A., “On Shunkov groups saturated with linear groups”, Sib. matem. zhurn., 57:3 (2016), 222–235 | MR | Zbl

[5] Senashov V. I., Shunkov V. P., Groups with limb conditions, SB RAS, Novosibirsk, 2001 | MR

[6] Cherep A. A., “On the set of elements of finite order in a biprimitively finite group”, Algebra i logika, 26:4 (1987), 518–521 | MR

[7] Shlepkin A. A., “Shunkov groups saturated with linear and unitary groups of degree 3 over fields of odd orders”, Sib. elektron. matem. izv., 13 (2016), 341–351 | MR | Zbl

[8] Шлёпкин А. А., “Периодические группы, насыщенные конечными простыми группами лиева типа ранга 1”, Алгебра и логика, 57:4 (2018), 118–125 | Zbl

[9] Shlepkin A. K., “On Certain Torsion Groups Saturated with Finite Simple Groups”, Siberian Adv. Math., 9:2 (1993), 100–108 | MR | MR | Zbl

[10] Shlepkin A. K., “On conjugate biprimitively finite groups with the condition of primary minimality”, Algebra i logika, 22 (1983), 226–231 | MR | Zbl

[11] Shlepkin A. A., “On the periodic part of the Shunkov group saturated with wreathed groups”, Tr. IMM UrO RAN, 24, no. 3, 2018, 281–285 | MR

[12] Shlepkin A. A., “On a Sufficient Condition for the Existence of a Periodic Part in the Shunkov Group”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 2017, no. 22, 90–105 | MR | Zbl

[13] Carter R. W., Simple groups of Lie type, Wiley and Sons, New York, 1972 | MR | Zbl

[14] Lytkina D. V., Shlepkin A. A., “Periodic groups saturated with the linear groups of degree 2 and the unitary groups of degree 3 over finite fields of odd characteristic”, Siberian Adv. Math., 28:3 (2018), 175–186 | DOI | MR | Zbl

[15] Shlepkin A. A., “On a sufficient condition when an infinite group is not simple”, Journal of SFU, mathematics and physics, 11:1 (2018), 103–107 | MR