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This is not true for infinite insoluble groups. Let $K$ be the direct product of countably infinitely many copies $\langle a_i,b_i \rangle$ ($i \in {\mathbb N}$) of $S_3 = \langle a,b \mid a^2=b^3=(a …

It's not too hard if you know Schur's Theorem, which says that, if $G$ is a group with $G/Z(G)$ finite, then $G'$ is finite. Let's assume that for the moment. We need to prove that, if $a$ and $b$ ar …

This seems surprisingly difficult! Let's try and do it by induction on the nilpotency class of $G$. The result is clear for abelian groups. Let $Z$ be the last nontrivial subgroup in the lower centra …

In the free class 2 nilpotent group with generators $x,y$, we have, for example, $xy^2x=yx^2y$. More generally, since nilpotent groups have polynomial growth, they cannot contain free semigroups of r …

$$a^ic^j = c^j(ab^j)^i.$$ I expect you could use your existing formula for $a^ib^j$ to write $(ab^j)^i$ in the form $b^ka^l$, but I will leave that to you!