Determine the number of abelian groups of order \(200\) up to isomorphism, and list all isomorphism classes in invariant factor and elementary divisor forms.
Solution
Since \[ 200=2^3 5^2, \] a finite abelian group of order \(200\) is the product of an abelian group of order \(2^3\) and one of order \(5^2\). The partitions of \(3\) are \(3\), \(2+1\), and \(1+1+1\); the partitions of \(2\) are \(2\) and \(1+1\). Hence there are \[ p(3)p(2)=3\cdot2=6 \] isomorphism classes.
Their elementary-divisor forms are \[ \begin{aligned} &\mathbb Z_8\times\mathbb Z_{25},\\ &\mathbb Z_8\times\mathbb Z_5\times\mathbb Z_5,\\ &\mathbb Z_4\times\mathbb Z_2\times\mathbb Z_{25},\\ &\mathbb Z_4\times\mathbb Z_2\times\mathbb Z_5\times\mathbb Z_5,\\ &\mathbb Z_2^3\times\mathbb Z_{25},\\ &\mathbb Z_2^3\times\mathbb Z_5^2. \end{aligned} \]
Combining relatively prime primary factors gives the corresponding invariant-factor forms \[ \begin{aligned} &\mathbb Z_{200},\\ &\mathbb Z_5\times\mathbb Z_{40},\\ &\mathbb Z_2\times\mathbb Z_{100},\\ &\mathbb Z_{10}\times\mathbb Z_{20},\\ &\mathbb Z_2\times\mathbb Z_2\times\mathbb Z_{50},\\ &\mathbb Z_2\times\mathbb Z_{10}\times\mathbb Z_{10}. \end{aligned} \] In each invariant-factor decomposition, each factor divides the next.