Free central extensions of groups and modular Lie powers of relation modules

The most prominent special case of our main result is that the free centre-by-(nilpotent of class (c - 1))-by-abelian groups F/[γc(F'), F] are torsion-free whenever c is divisible by at least two distinct primes. This is in stark contrast to the case where c is a prime or c = 4, where these relatively free groups contain non-trivial elements of finite order. © 2010 American Mathematical Society.