Geodesic
Two questions in the Kourovka Notebook
Algebra i logika, Tome 52 (2013) no. 5, pp. 632-637

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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. 
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     author = {A. I. Sozutov and E. B. Durakov},
     title = {Two questions in the {Kourovka} {Notebook}},
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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/