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The most prominent special case of our main result is that the free centre-by-(nilpotent of class (c-1))-by-abeliangroups  F/[\gamma_c(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

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