Periodic not implies locally finite

Statement
The statement can be made in two equivalent ways:


 * 1) It is possible to have a group $$G$$ that is both finitely generated and periodic (i.e., every element of $$G$$ has finite order), but is not locally finite.
 * 2) it is possible to have a periodic group $$G$$ that is not a locally finite group.

Proof
We will give examples of (1). Any example of (1) will automatically be an example of (2).

Grigorchuk group
The Grigorchuk group is a finitely generated periodic group: in fact, it is a 2-generated 2-group: it has a generating set of size two, and the order of every element is a power of 2. However, the group is not finite. In fact, it is a just infinite group.

Note that although the Grigorchuk group is periodic, it is not a group of finite exponent.

Tarski monsters and more generally negative answers to Burnside's question
The Burnside problem asks whether the group $$B(d,n)$$ is finite, where $$B(d,n)$$ is the quotient of the free group on $$d$$ generators by the relation that the $$n^{th}$$ power of every element is the identity. If the answer is "no" so that the Burnside group is infinite, then $$B(d,n)$$ gives an example of a finitely generated periodic group -- in fact, one of finite exponent-- that is not finite.

The most well known such examples are the Tarski monsters, which exist for all sufficiently large prime values of $$n$$ with $$d = 2$$.