# Math Help - Factor group of an external direct product

1. ## Factor group of an external direct product

Find the order of $Z \oplus Z | <(2,2)>$

Now, the elements in $Z \oplus Z$ are (0,0) and (1,1), so the order of it is 2. but I'm confuse about the second one, what is the order of that?

2. Originally Posted by tttcomrader
Find the order of $Z \oplus Z | <(2,2)>$

Now, the elements in $Z \oplus Z$ are (0,0) and (1,1), so the order of it is 2. but I'm confuse about the second one, what is the order of that?
Here $\mathbb{Z}\oplus \mathbb{Z} = \{... ,(-2,-2),(-1,-1),(0,0),(1,1),(2,2),...\}$
Write out $\left<(2,2)\right>$ which means $\{...,(-4,-4),(-2,-2),(0,0),(2,2),(4,4),...\}$.
Now notice that,
$\left< (2,2) \right> \mbox{ and }(1,1) + \left< (2,2)\right>$.
Form a complete list of co-sets, i.e. the even ones and the odd ones.
Thus, this factor group is of order 2.