# Thread: Isomorphism of abelian groups

1. ## Isomorphism of abelian groups

Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b)

Show that

$\displaystyle Z_{a} \oplus Z_{b}$ is isomorphic to $\displaystyle Z_{x} \oplus Z_{y}$.

2. Originally Posted by aliceinwonderland
Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b)

Show that

$\displaystyle Z_{a} \oplus Z_{b}$ is isomorphic to $\displaystyle Z_{x} \oplus Z_{y}$.
Use prime power decompositions.
If $\displaystyle N = \prod_{j}p_j^{a_j}$ then $\displaystyle \mathbb{Z}_N \simeq \bigoplus_j \mathbb{Z}_{p_j^{a_j}}$