TITLE: Does having a finite number of generators, each having finite order, imply that the group is finite?
QUESTION [22 upvotes]: Suppose that I have a group $G$ with a finite number of generators $(a_1,a_2,\ldots,a_n)$, and the orders $|a_1|,|a_2|,\ldots,|a_n|$ are all finite. Does it follow that the order of the group $|G|$ is necessarily finite?
I'm not sure how to even approach proving/disproving this problem. To try disproving it, I tried finding an infinite group that had a finite number of generators each with finite order, but everything I tried didn't seem to work:

$\mathbb{Z}$ has generators of infinite order
$GL_n(\mathbb{R})$ the set of invertible $n\times n$ matrices, also has generators of infinite order
The infinite cyclic group has a finite number of generators, but of infinite order

I don't really know many examples of groups (infinite or finite), so I got bummered here.

REPLY [30 votes]: In plane Euclidean geometry, the group generated two reflections in two parallel lines provides an easy example. The product of the two reflections is a nonzero translation, which has infinite order.