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$2$-group; locally finite group; normal-by-finite subgroup; core-finite group
A group $G$ has all of its subgroups normal-by-finite if $H/H_{G}$ is finite for all subgroups $H$ of $G$. The Tarski-groups provide examples of $p$-groups ($p$ a ``large'' prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$-group with every subgroup normal-by-finite is locally finite. We also prove that if $| H/H_{G} | \leq 2$ for every subgroup $H$ of $G$, then $G$ contains an Abelian subgroup of index at most $8$.
[1] Buckley, J. T., Lennox, J. C., Neumann, B. H., Smith, H., Wiegold, J.: Groups with all subgroups normal-by-finite. J. Aust. Math. Soc., Ser. A 59 (1995), 384-398. DOI 10.1017/S1446788700037289 | MR 1355229 | Zbl 0853.20023
[2] Cutolo, G., Khukhro, E. I., Lennox, J. C., Rinauro, S., Smith, H., Wiegold, J.: Locally finite groups all of whose subgroups are boundedly finite over their cores. Bull. Lond. Math. Soc. 29 (1997), 563-570. DOI 10.1112/S0024609397003068 | MR 1458716 | Zbl 0904.20030
[3] Kegel, O. H., Wehrfritz, B. A. F.: Locally Finite Groups. North-Holland Mathematical Library 3, North-Holland Publishing, Amsterdam (1973). MR 0470081 | Zbl 0259.20001
[4] Lennox, J. C., Hassanabadi, A. Mohammadi, Stewart, A. G. R., Wiegold, J.: Nilpotent extensibility and centralizers in infinite 2-groups. Proceedings of the Second International Group Theory Conference (Bressanone, 1989) Rend. Circ. Mat. Palermo (2) Suppl. No. 23 (1990), 209-219. MR 1068362 | Zbl 0705.20033
[5] Ol'shanskiĭ, A. Yu.: Geometry of Defining Relations in Groups. Mathematics and Its Applications. Soviet Series 70, Kluwer Academic Publishers, Dordrecht (1991). DOI 10.1007/978-94-011-3618-1 | MR 1191619 | Zbl 0732.20019
[6] Robinson, D. J. S.: A Course in the Theory of Groups. Graduate Texts in Mathematics 80, Springer, New York (1996). DOI 10.1007/978-1-4419-8594-1 | MR 1357169 | Zbl 0836.20001
[7] Wilkens, B.: More on core-2 2-groups. J. Group Theory 20 (2017), 193-225. DOI 10.1515/jgth-2016-0035 | MR 3619126 | Zbl 1370.20017
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