A combinatorial identity for the p-binomial coefficient based on abelian groups

For a non-negative integer $k$ and a positive integer $n$ with $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian groups. A purely combinatorial proof is not known for this identity. While proving this identity, for $r,s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underline{\lambda}$ a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to the combinatorial formula for subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise many enumeration formulae which are polynomial in $p$ with non-negative integer coefficients.