A group *G* is called * periodic* if every element has finite order; in other words, for each *g* in *G*, there exists some positive integer *n* such that *g*^{n} = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the *p*^{∞}-group which are infinite periodic groups; but the latter group cannot be finitely generated.

The **general Burnside problem** can be posed as: if *G* is a periodic group, and *G* is finitely generated, then is *G* necessarily a finite group?

This question was answered in the negative in 1964, when it was shown that there exists an infinite *p*-group which can be finitely generated.

As a related question which *seems* as if it it might have an easier answer, consider a periodic group *G* with the additional property that there exists a single integer *n* such that for all *g* in *G*, *g*^{n} = 1. A group with this property is said to be *periodic with bounded exponent* *n*, or just a *group with exponent* *n*.

Then the **Burnside problem** is stated as, if *G* is a finitely generated group with exponent *n*, is *G* finite?

This problem also has a negative answer, as was shown by an example due to S.I. Adan and P.S. Novikov
in 1968; a more famous class of counterexamples (given in 1982) are the Tarski Monsters - finitely generated infinite groups where every subgroup is a cyclic group of order *p*, where *p* is a prime greater than 10^{75}.
The problem of completely determining for which particular exponents *n* the answer to the Burnside Problem is in the positive has turned out to be more intractable.

To summarize the results to date, let *F*_{r} be the free group of rank *r*; and given a fixed integer *n*, let *F*_{r}^{n} be the subgroup of *F*_{r} generated by the set {*g*^{n} : *g* in *F*_{r}}. *F*_{r}^{n} is a normal subgroup of *F*_{r}; since if *h* = *a*_{1}^{n}*a*_{2}^{n}...*a*_{m}^{n} is in *F*_{r}^{n}, then

*g*^{ -1}*hg*= (*g*^{ -1}*a*_{1}^{n}*g*)(*g*^{ -1}*a*_{2}^{n}*g*)...(*g*^{ -1}*a*_{m}^{n}*g*) = (*g*^{ -1}*a*_{1}*g*)^{n}(*g*^{ -1}*a*_{2}*g*)^{n}...(*g*^{ -1}*a*_{m}*g*)^{n}

We then define the *Burnside free group* B(*r*, *n*) to be the factor group *F*_{r}/(*F*_{r}^{n}).

If *G* is any finitely generated group of exponent *n*, then *G* has a presentation including relations {*g*^{n} = 1} for all *g* in *G*, plus some additional relations. *G* is then a homomorphic image of B(*r*, *n*) for some *r*; so the Burnside problem can be re-stated as: for which positive integers *r*, *n* is B(*r*,*n*) finite?

Burnside proved some easy cases in his original paper:

- Clearly if
*r*= 1, then for all*n*, B(1,*n*) =*C*_{n}, the cyclic group of order*n*. - B(
*r*, 2) is the direct product of*r*copies of*C*_{2}.- This follows since, for any
*a*,*b*in B(*r*, 2), we have that (*ab*)(*ab*) = 1, and*a*=*a*^{ -1}; therefore*ab*=*ba*, and so B(*r*,2) is abelian, and so every element of B(*r*,2) is of the form*a*_{1}^{n1}*a*_{2}^{n2}...*a*_{r}^{nr}, where*n*_{i}is either 0 or 1 and {*a*_{i}} is the set of*r*generators.

- This follows since, for any

One hundred years later, the following additional results have been established:

- B(
*r*,3), B(*r*,4), and B(*r*,6) are finite for all*r*. - B(
*r*,*n*) is infinite if*r*> 2 and*n*> 12.

The **restricted Burnside problem** (formulated in the 1930s) asks another related question: are there only *finitely many* finite *r*-generator groups of exponent *n*? (An *r*-generator group is group which can be generated by *r* elements.)

If this holds for a given *r* and *n*, then consider subgroups *H* and *K* of B(*r*, *n*), where both *H* and *K* have finite index. The intersection of *H* and *K* then also has finite index. Let *M* be the intersection of *all* subgroups of B(*r*, *n*) which have finite index. *M* is a normal subgroup of B(*r*, *n*) (otherwise, there exists a subgroup *g*^{ -1}*Mg* with finite index containing elements not in *M*). We can then define B_{0}(*r*,*n*) to be the factor group formed by B(*r*,*n*)/*M*. B_{0}(*r*,*n*) is a finite group; and every finite *r*-generator group of exponent *n* is a homomorphic image of B_{0}(*r*,*n*).

The restricted Burnside problem was answered in the affirmative by Efim Zelmanov, for which he was awarded the Fields Medal in 1994.