Geodesic
Three S. N. Chernikov's questions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 23-25 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The author shows: the class of all periodic non-locally finite and non-locally nilpotent $FNN$-groups is non-empty and wide; an arbitrary binary graded $\overline{IH}$-group is solvable. At the same time, the author solves three natural S. N. Chernikov's questions. Also the author establishes that a non-Chernikov non-abelian group with normal such subgroups is solvable iff it is binary graded.
Keywords: a locally nilpotent group, a locally finite group, a locally graded group, a binary graded group.
Mots-clés : an $FNN$-group 
@article{TIMM_2012_18_3_a2,
     author = {N. S. Chernikov},
     title = {Three {S.} {N.~Chernikov's} questions},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {23--25},
     year = {2012},
     volume = {18},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a2/}
}
TY  - JOUR
AU  - N. S. Chernikov
TI  - Three S. N. Chernikov's questions
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2012
SP  - 23
EP  - 25
VL  - 18
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a2/
LA  - en
ID  - TIMM_2012_18_3_a2
ER  - 
%0 Journal Article
%A N. S. Chernikov
%T Three S. N. Chernikov's questions
%J Trudy Instituta matematiki i mehaniki
%D 2012
%P 23-25
%V 18
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a2/
%G en
%F TIMM_2012_18_3_a2
N. S. Chernikov. Three S. N. Chernikov's questions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 23-25. http://geodesic.mathdoc.fr/item/TIMM_2012_18_3_a2/

[1] Chernikov S. N., “An investigation of groups with prescribed properties of their subgroups”, Ukrain. Mat. Z., 21:2 (1969), 193–209 (in Russian) | MR | Zbl

[2] Ivanov S. V., “Three theorems on the imbedding of groups”, XI All-Union symp. theory groups (Kungurka, 31.01–02.02.1989), Thes. talks, Ural Branch Acad. Sci. USSR, Sverdlovsk, 1989, 51–55 (in Russian)

[3] Ivanov S. V., Olshanskii A. Yu., “Some applications of graded diagrams in combinatorial group theory”, London Math. Soc. Lect. Note Ser., 160, 1991, 258–308 | MR | Zbl

[4] Dedekind R., “Über Gruppen, deren sämmtliche Teiler Normalteiler sind”, Math. Ann., 48 (1897), 548–561 | DOI | MR | Zbl

[5] Baer R., “Situation der Untergruppen und Structur der Gruppe”, Sitz. Ber. Heidelberg Akad., 2 (1933), 12–17 | Zbl

[6] Chernikov S. N., “Infinite non-abelian groups with the condition of invariance for infinite non-abelian subgroups”, Dokl. Akad. Nauk SSSR, 194:6 (1970), 1280–1283 (in Russian) | MR | Zbl

[7] Chernikov S. N., “Infinite non-abelian groups where all the infinite non-abelian subgroups are invariant”, Ukrain. Mat. Z., 23:5 (1971), 604–628 (in Russian) | MR | Zbl

[8] Chernikov S. N., Groups with prescribed properties of the system of subgroups, Nauka, Moskva, 1980, 384 pp. (in Russian) | MR

[9] Olshanskii A. Yu., “Infinite groups with cyclic subgroups”, Dokl. Akad. Nauk SSSR, 245:4 (1979), 785–787 (in Russian) | MR

[10] Olshanskii A. Yu., “An infinite group with subgroups of prime order”, Izvestija Akad. Nauk SSSR. Ser. Mat., 44:2 (1980), 309–321 (in Russian) | MR | Zbl

[11] Olshanskii A. Yu., Geometry of defining relations in groups, Nauka, Moskva, 1989, 448 pp. (in Russian) | MR

[12] Chernikov S. N., “Infinite groups defining by properties of the system of infinite subgroups”, VI All-Union symp. theory groups, Coll. works, Nauk. dumka, Kiev, 1980, 5–22 (in Russian) | MR

[13] Kargapolov M. I., Merzljakov Yu. I., Fundamentals of the theory of groups, Nauka, Moskva, 1972, 240 pp. (in Russian) | Zbl

[14] Chernikov N. S., “Groups with minimal conditions”, Fundamental. i prykl. matematika, 14:5 (2008), 219–235 (in Russian) | MR

[15] Chernikov N. S., “Groups with minimal conditions”, J. Math. Sci., 163:6 (2009), 774–784 | DOI | MR

[16] Miller G. A., Moreno H. C., “Non-abelian groups in which every subgroup is abelian”, Trans. Amer. Math. Soc., 4:4 (1903), 398–404 | DOI | MR | Zbl