Is there a number $n$ such that there are exactly 1 million abelian groups of order $n$?
Can anyone please explain. I would yes because numbers are infinitive, and so any number n can be expressed as a direct product of cyclic groups of order n.
Can anyone please help me understand.
Thank you.
Best Answer
By the fundamental theorem for finite abelian groups the number of abelian groups of order $n=p_1^{n_1}\dots p_k^{n_k}$ is the product of the partition numbers of $n_i$.
Note that the partition number of $2$ is $2$ and the partition number of $4$ is $5$. Since $10^6=2^6\cdot 5^6$ such an $n$ therefore exists.