The Hall square in finite abelian p-groups

Kernels and cokernels of homomorphisms between finite abelian $p$-groups are finite abelian $p$-groups, and finite direct sums remain finite abelian $p$-groups. Transporting each kernel or cokernel along a bijection of its underlying finite set with a standard set gives the required object in the chosen model, with the same universal property. Thus the model is abelian using [F1]. For fixed finite groups $M,N$, Hom is a finite set. Any short exact sequence with endpoints $M,N$ has a middle group of cardinality $\mathrm{\mid }M\mathrm{\mid }\mathrm{\mid }N\mathrm{\mid }$; there are finitely many group laws on a finite set of that size and finitely many endpoint maps. Therefore its Yoneda extension set is finite. The stated group-table model is a set of objects, so [F3] applies.

If $0\to C_p\to L\to C_p\to 0$ is exact, then $\mathrm{\mid }L\mathrm{\mid }=p^2$. If $L$ contains an element of order $p^2$, then $L\cong C_{p^2}$. Otherwise every nonzero element has order $p$, so scalar multiplication by ${\mathbb{F}}_p$ makes $L$ a two-dimensional vector space and $L\cong C_p\oplus C_p$. Thus these are the only possible middle groups.

In $C_{p^2}$, the unique subgroup of order $p$ is $p\mathbb{Z}\mathrm{/}{p}^2\mathbb{Z}$, and its quotient is $C_p$, giving coefficient one by [F2]. In $C_p\oplus C_p$, every one-dimensional subspace is a subgroup isomorphic to $C_p$, and its quotient is $C_p$. The number of such subspaces is $(p^2-1)\mathrm{/}(p-1)=p+1$, by partitioning nonzero vectors into lines.

The extension classification and counts of step 2.1 give the formula. Moreover $\mathrm{End}(C_{p^2})\cong \mathbb{Z}\mathrm{/}{p}^2\mathbb{Z}$ additively, which has an element of order $p^2$; it is not an ${\mathbb{F}}_p$-vector space. Thus the category is not ${\mathbb{F}}_p$-linear.