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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.
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
Linking options:
https://www.mathnet.ru/eng/al608 https://www.mathnet.ru/eng/al/v52/i5/p632
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