Group generators and isomoprhisms Hello I have a rather quick question about the following;
I want to show that the groups
$\mathbb{Z_{2}} \times \mathbb{Z_{3}} \cong \mathbb{Z_{6}}$
Two questions;
I learnt the Chinese remainder theorem for rings, does it also hold for groups?
Second question
I know that any two cyclic groups with the same number of elements are isomorphic,
is the following a valid understanding of generators etc,
from $(1,1)$ since we are dealing with $\mathbb{Z}$ then our operations are addition (as 'multiplication') from addition and multiplication we can form all other 6 ordered pairs of such and therefore is that sufficient to conclude it is a cyclic group with 6 elements and clearly then isomorphic to $\mathbb{Z}$

I don't know about the Chinese Remainder Theorem for groups.  But as for your second question, it is sufficient to show that $\mathbb{Z}_2 \times \mathbb{Z}_3$ is cyclic by demonstrating that $(1,1)$ is a generator.
