I'm stuck on this one, could use a hand:
Let $c$ be the nilpotency rank of a nilpotent group $H$ (the smallest $c$ such that $H$ has a central series of length $c$ is called the nilpotency class of $H$).
Show that for each $h\in H$:
$\langle h,[H,H] \rangle$ is a nilpotent group of nilpotecy rank at most $c-1$.
(Where $[H,H] = \{h_1^{-1}h_2^{-1}h_1h_2 | h_1,h_2 \in H\}$ and $\langle a,b \rangle$ is the group generated by the elements a,b).
Thank you!
This follows from the definition of the nilpotency class of $H$. Let $\gamma_1(H)=[H,H]$, $\gamma_{i+1}=[H,\gamma_i(H)]$ for all $i\ge 1$, then $H$ has nilpotency class $c$ if $\gamma_c(H)\neq 0$ and $\gamma_{c+1}(H)=0$. If $H$ has nilpotency class $c$, then $\langle h,[H,H]\rangle \subseteq [H,[H,H]]=\gamma_2(H)$.