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Algebra Logika, 2013, Volume 52, Number 5, Pages 632–637 (Mi al608)  

This article is cited in 2 scientific papers (total in 2 papers)

Two questions in the Kourovka Notebook

A. I. Sozutovab, E. B. Durakova

a Siberian Federal University, pr. Svobodnyi 82, Krasnoyarsk, 660049, Russia
b Reshetnev Siberian State Aerospace University, pr. Gazety Krasnoyarskii Rabochii 31, Krasnoyarsk, 660037, Russia

Abstract: 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.

Full text: PDF file (122 kB)
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English version:
Algebra and Logic, 2013, 52:5, 422–425

Bibliographic databases:

UDC: 512.54
Received: 05.09.2013

Citation: A. I. Sozutov, E. B. Durakov, “Two questions in the Kourovka Notebook”, Algebra Logika, 52:5 (2013), 632–637; Algebra and Logic, 52:5 (2013), 422–425

Citation in format AMSBIB
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\by A.~I.~Sozutov, E.~B.~Durakov
\paper Two questions in the Kourovka Notebook
\jour Algebra Logika
\yr 2013
\vol 52
\issue 5
\pages 632--637
\mathnet{http://mi.mathnet.ru/al608}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184665}
\transl
\jour Algebra and Logic
\yr 2013
\vol 52
\issue 5
\pages 422--425
\crossref{https://doi.org/10.1007/s10469-013-9254-5}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84889052809}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. I. Sozutov, E. B. Durakov, “On groups with isolated involution”, Siberian Math. J., 55:4 (2014), 706–714  mathnet  crossref  mathscinet  isi
    2. A. I. Sozutov, “Groups with the quasicyclic centralizer of a finite involution”, Siberian Math. J., 57:5 (2016), 881–883  mathnet  crossref  crossref  isi  elib  elib
  • Алгебра и логика Algebra and Logic
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