Geodesic
Infinite groups of finite period
Algebra i logika, Tome 54 (2015) no. 2, pp. 243-251.

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

It is proved that there exist periodic groups containing an element of even order and only trivial normal $2$-subgroups in which every pair of involutions generates a $2$-group. This gives a negative answer to Question 11.11a in the Kourovka Notebook. Furthermore, we point out examples of finite simple groups that are recognizable by spectrum in the class of finite groups but not recognizable in the class of all groups.
Keywords: periodic group, periodic product, spectrum of group, recognizability by spectrum, Baire–Suzuki theorem, modular group. 
@article{AL_2015_54_2_a6,
     author = {V. D. Mazurov and A. Yu. Ol'shanskii and A. I. Sozutov},
     title = {Infinite groups of finite period},
     journal = {Algebra i logika},
     pages = {243--251},
     publisher = {mathdoc},
     volume = {54},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2015_54_2_a6/}
}
TY  - JOUR
AU  - V. D. Mazurov
AU  - A. Yu. Ol'shanskii
AU  - A. I. Sozutov
TI  - Infinite groups of finite period
JO  - Algebra i logika
PY  - 2015
SP  - 243
EP  - 251
VL  - 54
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2015_54_2_a6/
LA  - ru
ID  - AL_2015_54_2_a6
ER  - 
%0 Journal Article
%A V. D. Mazurov
%A A. Yu. Ol'shanskii
%A A. I. Sozutov
%T Infinite groups of finite period
%J Algebra i logika
%D 2015
%P 243-251
%V 54
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2015_54_2_a6/
%G ru
%F AL_2015_54_2_a6
V. D. Mazurov; A. Yu. Ol'shanskii; A. I. Sozutov. Infinite groups of finite period. Algebra i logika, Tome 54 (2015) no. 2, pp. 243-251. http://geodesic.mathdoc.fr/item/AL_2015_54_2_a6/

[1] Nereshënnye voprosy teorii grupp, Kourovskaya tetrad, 18-e izd., In-t matem. SO RAN, Novosibirsk, 2014 http://www.math.nsc.ru/~alglog/18kt.pdf

[2] D. M. Sonkin, On groups of large exponents n and n-periodic products, PhD Thesis, Vanderbilt Univ., Nashville, Tennessee, 2005, 83 pp. http://etd.library.vanderbilt.edu/available/etd-06032005-025224/ | MR

[3] S. I. Adyan, “Periodicheskie proizvedeniya grupp”, Teoriya chisel, matematicheskii analiz i ikh prilozheniya, Tr. MIAN SSSR, 142, Nauka, M., 1976, 3–21 | MR | Zbl

[4] S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Int. J. Algebra Comput., 4:1/2 (1994), 1–308 | DOI | MR | Zbl

[5] A. Kh. Zhurtov, V. D. Mazurov, “Raspoznavanie prostykh grupp $L_2(2^m)$ v klasse vsekh grupp”, Sib. matem. zh., 40:1 (1999), 75–78 | MR | Zbl

[6] D. V. Lytkina, A. A. Kuznetsov, “Recognizability by spectrum of the group $L_2(7)$ in the class of all groups”, Sib. elektron. matem. izv., 4 (2007), 136–140 | MR | Zbl

[7] A. S. Mamontov, E. Yabara, “Raspoznavanie gruppy $PSL3(4)$ po mnozhestvu poryadkov elementov v klasse vsekh grupp”, Algebra i logika (to appear)

[8] R. Brandl, W. J. Shi, “The characterization of $PSL(2,q)$ by its element orders”, J. Algebra, 163:1 (1994), 109–114 | DOI | MR | Zbl

[9] B. Huppert, Endliche Gruppen, v. I, Grundlehren mathem. Wiss. Einzeldarstel., 134, Springer-Verlag, Berlin a.o., 1979 | Zbl

[10] M. M. Postnikov, Teorema Ferma. Vvedenie v teoriyu algebraicheskikh chisel, Nauka, M., 1986 | MR

[11] C. K. Caldwell, G. L. Honaker (Jr.), Prime curios!, The dictionary of prime number trivia, CreateSpace, Martin, 2009

[12] E. Jabara, D. V. Lytkina, V. D. Mazurov, “Some groups of exponent 72”, J. Group Theory, 17:6 (2014), 947–955 | DOI | MR | Zbl