Burnside problem

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This article describes an open problem in the following area of/related to group theory: presentation theory


For what values of n are the following equivalent conditions true?

  1. Every group of exponent n is locally finite
  2. Every finitely generated group of exponent n is finite
  3. For every positive integer d, the Burnside group B(d,n) is a finite group.

Note that, at least prima facie, it is possible, for a given n, that B(d,n) is finite for small d but infinite for large d.


Small exponent cases

Value of n Answer to Burnside's question Nilpotency class of B(d,n) in terms of n (assume d \ge 1) Order of B(d,n) in terms of n,d Explanation and comments
1 Yes 0 1 The only possible group is the trivial group
2 Yes 1 2^d exponent two implies abelian, so any group of exponent 2 must be an elementary abelian 2-group
3 Yes 1 for d = 1
2 for d = 2
3 for d \ge 3
3^r where r = d + \binom{d}{2}+ \binom{d}{3} exponent three implies class three: this follows from exponent three implies 2-Engel for groups, 2-Engel implies class three for groups
4 Yes 1 for d = 1
5 Unknown 1 for d = 1
Unknown for d \ge 2
6 Yes 1 for d = 1
Not nilpotent for d \ge 2

Large exponent cases

Statement Best known bounds
For very large values of n, the group B(d,n) is infinite for every d > 1. In particular, B(2,n) is infinite. Definitely true for all primes greater than 10^{75}, because of the existence of Tarski monsters. This property also holds for all odd n greater than or equal to 665, and for all even n greater than or equal to 8000.

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