Number of Abelian Groups

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $n \in \Z_{\ge 1}$ be a (strictly) positive integer.

Let:

$n = \ds \prod_{i \mathop = 1}^s p_i^{m_i}$

where the $p_i$ are distinct primes.

Let $\map {\nu_a} n$ denote the number of abelian groups of order $n$.


Then:

$\map {\nu_a} n = \ds \prod_{i \mathop = 1}^s \map {\nu_a} {p_i^{m_i} }$

where:

$\map {\nu_a} {p_i^{m_i} }$ is the number of integer partitions of $m_i$.


Proof



Sources