# Burnside problem

**Conjecture**If a group has generators and exponent , is it necessarily finite?

It is possible to define the to be the group generated by with relations where ranges over every word in the generators. There is a universality property: Any homomorphism where has r generators and exponent dividing can be written as a composition of a homomorphism with a homomorphism . Some cases of this are known: is a cyclic group of order , for any positive integer . is trivial for any positive integer . is isomorphic to the Cartesian product of cyclic groups of order , for any positive integer . This is because the relations make it easy to prove that the generators commute. is a finite group, and its order is is a finite group, and its order is is a finite group for any positive integer . The order is known for up to : is known to be infinite for sufficiently large and odd , as well as and divisible by .

## Bibliography

* indicates original appearance(s) of problem.