Let $G$ a finite twogenerated $p$group in which lower and upper central series coincide. Clearly we obtain that the upper central series become strongly central, we have also that at least half of the members of the upper central series are abelian. Are there not immediate results that involve this kind of group?

I thought you might find this interesting:
Proof: Since $\Gamma$ has rank $k$, we can fix a generating set $x_1, \ldots, x_k$ for $\Gamma$. Consider the map $\varphi: \Gamma_i \to (\Gamma_{i+1}/\Gamma_{i+2})^k$ given by $g \mapsto ([x_1,g], [x_2, g], \ldots, [x_k,g] )$. This map is a homomorphism since in any group $[x,yz] = [x,z][x,y]^z$. Further, its kernel is $\Gamma_{i+1}$, as for any $g \in \Gamma_i \setminus \Gamma_{i+1}$, if $\varphi(g) = 1$, then $g$ is in the center of $\Gamma/\Gamma_{i+2}$. But this is impossible, as the center of $\Gamma/\Gamma_{i+2}$ is contained in $\Gamma_{i+1}/\Gamma_{i+2}$ since $\Gamma$ is a ULequivalent group. QED Some remarks:


