Finitely Presented Infinite Group has an Element of Infinite Order? Is it true that an infinite group which is finitely generated must have an element of infinite order? If not, I'd like a counterexample.
[Edit] 
Somewhat easily proven false.
More interesting (and as it happens, more relevant to my particular problem domain), what about finitely presented groups?

In full generality, this is false.  As mentioned in the comments, the Burnside groups can be infinite groups, finitely generated by construction, such that every element has bounded finite order. One can even force such a group to be simple.
There are positive results for certain families of groups, however.  Immediately I am reminded of solvable groups and hyperbolic groups.
Perhaps a much more interesting question is if there is a finitely presented, infinite group, with no element of infinite order.  AFAIK, this question is still wide open.
