21.150 (2026)

*No, not always. Let $A = \mathbb{F}_3[x, y]/I^3$ be the additive group of the quotient of the polynomial algebra $\mathbb{F}_3[x, y]$ by the ideal $I^3$, where $I$ is the ideal generated by $x, y$. Let $B$ be the group of automorphisms of $A$ generated by multiplication by $1 + x$ and $1 + y$. Let $G = A \rtimes B$. For $a = 1 \in A$ we have $C_B(a) = 1_B$. For $H = \langle a, B \rangle$ we have $Z(H) \cap H' = \langle x^2, xy, y^2 \rangle$, which has rank 3, while the rank of $B$ is 2 (P. Monticone, Letter of 21 March 2026; see also https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution_21_150-3.pdf).