Periodic not implies locally finite

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., periodic group) need not satisfy the second group property (i.e., locally finite group)
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Get more facts about periodic group|Get more facts about locally finite group
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated periodic group) need not satisfy the second group property (i.e., finite group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about finitely generated periodic group|Get more facts about finite group

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

Further information: 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.