Entropy of Cohen-Lenstra measures: the $u$-aspect

Let $\mathbf{H}({\nu }_{\mathrm{C}\mathrm{L}}^u)$ be the entropy of the Cohen-Lenstra measure on finite abelian $p$-groups associated to an integral unit-rank $0\le u\in \mathbb{N}$. In this note, we show that $0<\mathbf{H}({\nu }_{\mathrm{C}\mathrm{L}}^u)<\mathrm{\infty }$ for all $u$, $\mathbf{H}({\nu }_{\mathrm{C}\mathrm{L}}^u)$ is a strictly decreasing function of $u\ge 0$, and $\mathbf{H}({\nu }_{\mathrm{C}\mathrm{L}}^u)\stackrel{u\to \mathrm{\infty }}{\to }0$. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian $p$-groups.