# Isomorphic groups

by razorg425
Tags: groups, isomorphic
 P: 256 Start by proving that $$\mathbb{Z}_{21} \simeq \mathbb{Z}_3 \times \mathbb{Z}_7$$. What are the groups of units of $$\mathbb{Z}_3$$ and $$\mathbb{Z}_7$$? (Hint: it's important that GCD(3,7)=1.)
 P: 256 So we want to show that every element of $$\mathbb{Z}_3 \times \mathbb{Z}_7$$ is of the form $$n \cdot ( [1]_3, [1]_7 )$$. So for $$x, y \in \mathbb{Z}$$, we want $$( [x]_3, [y]_7 ) = n \cdot ( [1]_3, [1]_7 ) = ( [n]_3, [n]_7 )$$. So we need to find a number n so that $$x \equiv n \mod 3$$ and $$y \equiv n \mod 7$$. The Chinese Remainder Theorem is your friend.