Given an $n\times m$ matrix $A$ over a PID $R$, there exist $0\leq r\leq \min\{m,n\}$ and $e_1, \dotsc, e_r \in R\setminus\{0\}$ with $e_1\mid e_2\mid \cdots \mid e_r$ as well as matrices $U,V\in \textrm{Gl}_n(R)$ s.t. $$UAV = \begin{pmatrix} e_1 & & & | & \\ & \ddots & & |&\\ &&e_r & | & \\ \hline & & & | & 0&\\ \end{pmatrix}. $$ (This is the Smith normal form, with the $e_i$ being the elementary divisors of $A$.)

I'm interested in the change of the elementary divisors if we increase the size of our given matrix by a vector (i.e. $A \leftrightarrow (A\mid v)$ for some $v\in R^n$).

We know that the product of the first $l$ elementary divisors can not increase (w.r.t. divisibility), thus the amount of elementary divisors can not decrease. (Because $e_1 \cdots e_l$ is the greatest common divisor of all $l\times l$ minors.)

I would like to apply this to calculate the distribution of finite abelian $p$-groups as cokernels of $p$-adic matrices of size $n \times (n+1)$. The case for $n\times n$ matrices was already calculated by Friedman & Washington in On the distribution of divisor class groups of curves over a finite field. Their distribution converges to the Cohen-Lenstra distribution for $n\rightarrow \infty$, thus this would give me the distribution of finite abelian $p$-groups given by dividing a randomly (w.r.t. the Cohen-Lenstra distribution) drawn $p$-group by the subgroup generated by a random (w.r.t. the uniform distribution) element.



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