Partial-Burnside Groups

Abstract

We consider groups which have a presentation whose defining relators are all nth powers and in which every element has order dividing n, for a fixed odd n that is sufficiently large. Such groups are called Partial-Burnside groups. We examine subgroups of such groups, showin that every noncyclic subgroup of a Partial-Burnside group contains a noncyclic Partial-Burnside group. We show that the word problem is solvable if and only if the conjugacy problem is solvable. Additionally, we show that every finitely presented subgroup contains a noncyclic free group. Finally, we show that various product varieties are not finitely based. In particular, in joint work with Olshanskii, we show that BpBp is not finitely based for large primes p, answering a question of Gupta and Krasilnikov.