Economical adjunction of square roots to groups

Abstract

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

Yet another Freiheitssatz: Mating finite groups with locally indicable ones

“…The study of this notion has a long history, see, e.g., [1, 2, 4, 6–12, 15–18, 20–25, 27, 28, 32], and references therein. The following result is well known.…”