MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic?

For example, must they be solvable?

share|cite|improve this question
Which elements of the free group have noncyclic centralizers? – Goldstern Apr 26 '13 at 22:26
@Goldstern: Nonabelian free groups are not amenable. – Misha Apr 26 '13 at 23:22
@solovei: Now, this, is a reasonable question. For elementary amenable groups, the answer, I think, affirmative (i.e., all such groups are solvable). For nonelementary amenable groups, a comprehensive answer is well beyond the reach of the present technology. However, for the specific question about solvability, I would bet on a counterexample. For instance, I see nothing to prevent existence of a f.g. amenable group with trivial center, where every proper subgroup is cyclic (a version of Tarski monster). (If you ask A.Ol'shansky, he might be able to construct one for you.) – Misha Apr 26 '13 at 23:39
Sorry, I misread the question as "Can one characterize amenable groups BY the following property" (which would be silly). – Goldstern Apr 27 '13 at 0:24
Am I the only one who is bugged by questions that start in the title instead of being self-contained in the body? – Arturo Magidin Apr 27 '13 at 5:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.