Calculating number of abelian groups up to isomorphism

Question

$Problem$

Let p₁,......,p_n be distinct primes. Up to isomorphism ,how many Abelian groups are there of order p₁⁴p₂⁴....p_n⁴

$ Attempt$

Each p_i⁴ can be partitioned into 5 ways. So total number of Abelian groups is 5ⁿ

Is my approach correct ?

Answer 1

Consider $2^4$... You can have $\mathbb Z_{2^4},\mathbb Z_2×\mathbb Z_{2^3},\mathbb Z_2×\mathbb Z_2×\mathbb Z_{2^2},\mathbb Z_{2^2}×\mathbb Z_{2^2}$, or $\mathbb Z_2×\mathbb Z_2×\mathbb Z_2×\mathbb Z_2$... Thus you have $5$ different groups of order $2^4$.

Yes, the number of partitions of $4$ is apparently $5$: $4,1+3,1+1+2,2+2$ and $1+1+1+1$.

So you have $5^n$ such abelian groups.