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We first give a short group theoretic proof of the following result of Lackenby. If G is a large group, H is a finite index subgroup of G admitting an epimorphism onto a non-cyclic free group, and g 1 ,..., g k are elements of H, then the quotient of G by the normal subgroup generated by g n 1 ,..., g n k is large for all but finitely many n ∈ Z. In the second part of this note we use similar methods to show that for every infinite sequence of primes (p 1 , p 2 ,...), there exists an infinite finitely generated periodic group Q with descending normal series Q = Q 0 >Q 1 >..., such that ∩ i Q i = {1} and Q i-1 /Q i is either trivial or abelian of exponent p i .

Large groups and their periodic quotients