Re: [math-fun] Interesting unsolved problem in group theory

I learned from some recent postings on sci.math.research that if F(r) is a free group (on r >= 2 generators), and if G^n denotes the subgroup generated by all nth powers of elements of a group G, then it's unknown whether the quotient group,

B(r,n) = F(r) / F(r)^5 is finite or infinite.

It is known that B(r,n) is always finite for n = 2,3,4, and 6 (cf. Marshall Hall's book, "Group Theory"), as well as for sufficiently large n (apparently independent of r).

1992 S I Ivanov proved that B(r, n) is infinite for r > 2 and n >= 13. See A History of the Burnside Problem at http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Burnside_problem.html --Edwin