# Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite.

Given: Let $$G$$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements.

Show that $$G$$ is finite.

Also would love to know if it's possible to show that an infinite finitely generated nilpotent group has an infinite center.

• It would suffice to show $G'$ is also finitely-generated by torsion elements, since then $G/G'$ is finite (it is finitely generated and abelian) and $G'$ is finite by induction on nilpotence class. – runway44 Jun 13 at 0:43
• @runway44 Would love to know if you have a way to approach/solve that. – Ilan Aizelman WS Jun 15 at 12:12
• I would suggest using the lower central series $\gamma_i(G)$. If $\gamma_c(G)$ is the last nontrivial term, then it is central. By induction $G/\gamma_c(G)$ is finite, and you can then show that $\gamma_c(G) = [G,\gamma_{c-1}(G)]$ is finite. Similarly for the second problem, if $\gamma_c(G)$ is finite then by induction $G/\gamma_{c-1}(G)$ has infinite centre,$Z/\gamma_c(G)$,, and you can show that $|Z:Z \cap Z(G)|$ is fihite. – Derek Holt Jun 17 at 12:21
• You can also prove it using the results proved in detail in this post – Derek Holt Jun 17 at 12:32

The following argument is incomplete, although I did not realize it until after writing it out. In fact, I now see that it is a fleshing out of the comment by runway44 above. After further thought, the missing step (torsion generation of $$[G,G]$$) is quite substantial and indicates this is not the right approach. Instead, the argument outlined by Derek Holt appears to be the correct path.
The inductive hypothesis applied to $$[G,G]$$ implies that it is a finite group. Let $$g_1,\ldots,g_n$$ be a generating set of torsion elements for $$G$$. Then every $$g\in G$$ has a representation of the form $$g=h\prod_{i=1}^{n}g_i^{x_i},\qquad h\in [G,G],$$ where each $$x_i$$ ranges from $$0$$ to $$\textrm{ord}(g_i)-1$$. Indeed, we may iteratively all instances of $$g_1$$ to the end (keeping a commutator at the start of $$g$$), then $$g_2$$, and so on, always keeping track of the commutators incurred during the rearrangement in $$h$$.
From this representation, we obtain that $$|G|\leq \bigl|[G,G]\bigr| \prod_{i=1}^n\textrm{ord}(g_i)$$, in particular showing that $$G$$ is finite.