Combinatorial Conditions Forcing Commutativity of an Infinite Group

✍ Scribed by L. Brailovsky

• Book ID: 102569895
• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 271 KB
• Volume: 165
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1994.1119

✦ Synopsis

We show that the function (f(n)=\left\lceil\left(5 n^{2}-3 n-2\right) / 6\right\rceil) is the best possible squaring bound for infinite abelian groups. That is, if (G) is an infinite group and (k) is an integer (\geqslant 2), such that the condition, (\left|K^{2}\right| \leqslant f(k)), holds for every (k)-element subset (K \subseteq G) then (G) is abelian. Moreover, (f(n)) is the "maximal" integer valued function with this property. A characterization of central-by-finite groups appears in the proof. 1994 Academic Press, Inc.