# Burnside problem

 Importance: Outstanding ✭✭✭✭
 Author(s): Burnside, William
 Subject: Group Theory
 Keywords:
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 .
Burnside_problem Burnside Problem -- from Wolfram MathWorld