# Factors of the upper central series of a torsion-free nilpotent group are also torsion-free

Prove that the factors of the upper central series of a torsion-free nilpotent group are also torsion-free.

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This follows from induction and the fact that there is an injective homomorphism from $\gamma_{i+1}(G)/\gamma_{i}(G)$ into $\mathrm{Hom}(G/G',\gamma_{i}(G)/\gamma_{i-1}(G))$. The map sends the coset $g\gamma_i(G)$ to the homomorphism that sends $hG'$ to $[g,h]\gamma_{i-1}(G)$. –  user641 Jan 23 at 8:31
Ok, no problem. I supposed using the phrase "prove, show" (as in the excercise of the book) to be more "professional" as opposed to "I have been trying for hours, tried this-and-that, and cannot figure out how to..." The above exercise was needed to construct a counter-example where the upper central series is not equal to the lower central series. It suffices to find a LCS were the above result does not hold. Take therefore G = (1 nZ Z \\ 0 1 Z \\ 0 0 1) and [G,G] = (1 0 nZ \\ 0 1 0 \\ 0 0 1) such that $G/[G,G] \cong Z \oplus Z \oplus Z_m$ is not torsion-free. –  yannickvda Jan 23 at 10:59