TITLE: Non-abelian group with infinitely many abelian subgroups
QUESTION [12 upvotes]: I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?

REPLY [2 votes]: The set of $2\times 2$ matrices with real entries is non-Abelian when the operator is multiplication, but it has an infinite number of Abelian subgroups.
For example consider any subgroup of the form 
$$\{A | A = \begin{bmatrix} p^n & 0\\ 0 & 1\end{bmatrix} \mbox{ where } n\in \mathbb{Z}\}$$ 
where $p$ is a constant and can be any prime.