Abstract
Glauberman’s Z *-theorem [1] 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 [3, Question 11.13] and V. D. Mazurov [3, Question 17.71a].
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