Number of Abelian Groups
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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
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Sources
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.5$