Stack Exchange Network
Stack Exchange network consists of 181 Q&A communities including
Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this community
Anybody can ask a question
The best answers are voted up and rise to the top
11 years ago
Does every countable, infinite, amenable, periodic group $G$ contain an infinite locally finite subgroup?
I would be happy with an infinitely generated counterexample as long as it is countable.
A counterexample cannot be elementary amenable, as
elementary amenable periodic groups are locally finite.
First Grigorchuk's group is not a counterexample: it is a 2-group, and hence it contains an infinite abelian group (by a result of Held, "On abelian subgroups of infinite 2-groups").
Amenability of $G$ is essential, due to existence of Tarski monsters whose subgroups
are finite cyclic (constructed by Olshanskii).
Any infinite locally finite group contains an infinite abelian subgroup
(see a paper of Hall-Kulatilaka
here, so it is equivalent to ask whether $G$ contains an infinite abelian subgroup.
This question was asked
here in 2008, so perhaps it is an open problem.
Jun 1, 2012 at 15:00
Igor Belegradek Igor Belegradek
27.9k 2 2 gold badges 75 75 silver badges 171 171 bronze badges
@Ashot, Grigorchuk $p$-groups are branch, so one can easily construct infinite abelian subgroups as well. As generators one can just take elements from rigid stabilizers of different vertices, such that none of these vertices is a prefix of the other. So these groups don't provide counterexamples.
Jun 2, 2012 at 12:04
log in to answer this question.
Not the answer you're looking for? Browse other questions tagged
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our
Accept all cookies
Necessary cookies only