Submission date: 27 February 2017

Abstract:

The role of finite centralizers of involutions in pseudo-finite groups is analyzed. Using basic techniques from infinite group theory, it is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, an alternative proof of the corresponding result for periodic groups due to Hartley and Meixner is given, as well as a gently improvement regarding definable properties. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup, and that in any pseudo-finite group in which the centralizer of any element is finite or has finite index, the FC-center is a finite index definable subgroup.

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Full text arXiv 1702.08173: pdf, ps.