V.I. Zenkov. On intersections of nilpotent subgroups in finite groups with socle $L_2(2^m)\times L_2(2^n)$ ... P.126-134

In Theorem 1, it is proved for a finite group $G$ with socle $L_2(2^m)\times L_2(2^n)$ and nilpotent subgroups $A$ and $B$ that the condition $\min_G(A,B)\ne 1$ implies that $n=m=2$ and the subgroups $A$ and $B$ are $2$-groups. Here the subgroup $\min_G(A,B)$ is generated by smallest-order intersections of the form $A\cap B^g$, $g\in G$, and the subgroup $\mathrm{Min}_G(A,B)$ is generated by all intersections of the form $A\cap B^g$, $g\in G$, that are minimal with respect to inclusion. In Theorem 2, for a finite group $G$ with socle $A_5\times A_5$ and a Sylow 2-subgroup $S$, we give a description of the subgroups $\min_G(S,S)$ and $\mathrm{Min}_G(S,S)$. On the basis of Theorem 2, in Theorem 3 for a finite group $G$ with socle $A_5\times A_5$ we describe up to conjugation all pairs of nilpotent subgroups $(A,B)$ of $G$ for which $\min_G(A,B)\ne 1$.

Keywords: finite groups, nilpotent subgroup, intersection of subgroups

Received July 03, 2018

Revised October 24, 2018

Accepted October 29, 2018

Funding Agency: This work was supported by by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Viktor Ivanovich Zenkov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990, Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: v1i9z52@mail.ru

REFERENCES

1.   Burnside W. On groups of order $p^{\alpha}q^{\beta}$. Proc. London Math. Soc., 1904, vol. 2, no. 1, pp. 388–392.

2.   Burnside W. On groups of order $p^{\alpha}q^{\beta}$ (second paper), Proc. London Math. Soc., 1905, vol. 2, no. 2, pp. 432–437.

3.   Monakhov V.S. Normal subgroups of biprimary groups. Math. Notes, 1975, vol. 18, no. 6, pp. 877–886.

4.   Kabanov V.V.., Kondrat’ev A.S. Silovskie 2-podgruppy konechnykh grupp [Sylow 2-subgroups of finite groups]. Sverdlovsk: UrO RAN publ., 1979, 155 p.

5.   Zenkov V.I. Intersections of nilpotent subgroups in finite groups. Fundament. Prikl. Mat., 1996, vol. 2, no. 1, pp. 1–92 (in Russian).

6.   Zenkov V.I. On intersections of primary subgroups in the group Aut($L_n(2)$). Proc. Steklov Inst. Math., 2016, vol. 293, Suppl. 1, pp. 270–277. doi: 10.1134/S0081543816050230

7.   Zenkov V.I., Nuzhin Ya.N. On intersection of primary subgroups of odd order in finite almost simple groups. J. Math. Sci., 2017, vol. 221, no. 3, pp. 384–390. doi: 10.1007/s10958-017-3232-8

8.   Zenkov V.I. On intersections of nilpotent subgroups in finite symmetric and alternating groups. Proc. Steklov Inst. Math., 2014, vol. 285, Suppl. 1, pp. 203–208. doi: 10.1134/S0081543814050228

9.   Convay J.H. [et. al.]. Atlas of finite group. Oxford: Clarendon Press, 1985, 252 p. ISBN: 0-19-853199-0

10.   Gorenstein D., Lyons R. The local structure of finite groups of characteristic 2 type. Mem. Amer. Math. Soc., 1983, vol. 42, 731 p. ISBN: 0-8218-2276-4

11.   Zenkov V.I. Intersection of Abelian subgroups in finite groups. Math Notes, 1994, vol. 56, no. 2, pp. 869–871. doi: 10.1007/BF02110750

12.   Jamali A.R., Viseh M. On nilpotent subgroups containing nontrivial normal subgroups. J. Group Theory, 2010, vol. 13, no. 4, pp. 411–416. doi: 10.1515/jgt.2009.058

13.   Zenkov V.I., Mazurov V.D. On intersections of Sylow p-subgroups in finite groups. Algebra Logic, 1996, vol. 35, no. 4, pp. 236–240.

14.   Zenkov V.I. Intersections of two nilpotent subgroups in finite groups with socle $L_2(q)$. Sib. Math. J., 2016, vol. 57, no. 6, pp. 1002–1010. doi: 10.1134/S0037446616060070

15.   Kargapolov M.I., Merzljakov Ju.I. Fundamentals of the theory of groups. N Y; Heidelberg; Berlin: Springer-Verlag, 1979, Ser. Graduate Texts Math, 62, 203 p. ISBN: 978-1-4612-9966-0 . Original Russian text (1st ed.) published in Kargapolov M.I., Merzlyakov Yu.I. Osnovy teorii grupp. Moscow, Nauka Publ., 1972, 239 p.

16.   Busarkin V.M., Gorchakov Yu.M. Konechnye rascheplyaemye gruppy [Finite groups that admit partitions]. Moscow, Nauka Publ., 1968, 113 p.