Geodesic
Two questions in the Kourovka Notebook
Algebra i logika, Tome 52 (2013) no. 5, pp. 632-637
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

G. Glauberman's $Z^*$-theorem [J. Algebra, 4, No. 3, 403–420 (1966)] and the theorem of Bender are two most important tools for local analysis in the theory of finite groups. The $Z^*$-theorem generalizes the known Burnside and Brauer–Suzuki theorems on finite groups with cyclic and quaternion Sylow $2$-subgroups. Whether these theorems are valid in a class of periodic groups is unknown. We prove that the $Z^*$-theorem is invalid in the class of all periodic groups. In particular, this gives negative answers to questions of A. V. Borovik and V. D. Mazurov [see Unsolved Problems in Group Theory, The Kourovka Notebook, Questions 11.13 and 17.71a].
Keywords: finite group, Glauberman's $Z^*$-theorem. 
@article{AL_2013_52_5_a7,
     author = {A. I. Sozutov and E. B. Durakov},
     title = {Two questions in the {Kourovka} {Notebook}},
     journal = {Algebra i logika},
     pages = {632--637},
     year = {2013},
     volume = {52},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2013_52_5_a7/}
}
TY  - JOUR
AU  - A. I. Sozutov
AU  - E. B. Durakov
TI  - Two questions in the Kourovka Notebook
JO  - Algebra i logika
PY  - 2013
SP  - 632
EP  - 637
VL  - 52
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/AL_2013_52_5_a7/
LA  - ru
ID  - AL_2013_52_5_a7
ER  - 
%0 Journal Article
%A A. I. Sozutov
%A E. B. Durakov
%T Two questions in the Kourovka Notebook
%J Algebra i logika
%D 2013
%P 632-637
%V 52
%N 5
%U http://geodesic.mathdoc.fr/item/AL_2013_52_5_a7/
%G ru
%F AL_2013_52_5_a7
A. I. Sozutov; E. B. Durakov. Two questions in the Kourovka Notebook. Algebra i logika, Tome 52 (2013) no. 5, pp. 632-637. http://geodesic.mathdoc.fr/item/AL_2013_52_5_a7/

[1] G. Glauberman, “Central elements in core-free groups”, J. Algebra, 4:3 (1966), 403–420 | DOI | MR | Zbl

[2] D. Gorenstein, Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu, Mir, M., 1985 | MR | Zbl

[3] Nereshënnye voprosy teorii grupp. Kourovskaya tetrad, 17-e izd., In-t matem. SO RAN, Novosibirsk, 2010 http://www.math.nsc.ru/~alglog/17kt.pdf

[4] S. I. Adyan, Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl

[5] A. Yu. Olshanskii, Geometriya opredelyayuschikh sootnoshenii v gruppakh, Nauka, M., 1989 | MR

[6] A. I. Sozutov, N. M. Suchkov, N. G. Suchkova, Beskonechnye gruppy s involyutsiyami, Sib. federal. un-t, Krasnoyarsk, 2011