Theorem : The number of non-isomorphic abelian groups of order $p^n$, $p$ is a prime equals to number of partitions of $n$
From above theorem it implies that number of non isomorphic abelian groups of order $2^4$ equals to that of order $3^4$ and so on
That looks very strange to me. How this is possible. Please explain?
Presume you know the Fundamental theorem of finitely generated Abelian groups (https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Classification)? This statement is a direct consequence. For example, the partitions of $n=4$ into $4$, $3+1$, $2+2$, $2+1+1$ and $1+1+1+1$ give the non-isomorphic groups of order $p^4$: $\mathbb Z_{p^4}$, $\mathbb Z_{p^3}\oplus \mathbb Z_p$, $\mathbb Z_{p^2}\oplus \mathbb Z_{p^2}$, $\mathbb Z_{p^2}\oplus \mathbb Z_p\oplus \mathbb Z_p$ and $\mathbb Z_p\oplus \mathbb Z_p\oplus \mathbb Z_p\oplus \mathbb Z_p$.