Order of a factor group and whether it is cyclic

Question

Question:

Determine the order of $\left ( \mathbb{Z} \bigoplus \mathbb{Z}\right )/\left \langle \left ( 2,2 \right ) \right \rangle$.Is it cyclic?

The elements in the factor group are of the form $\left ( a,b \right )+\left \langle \left ( 2,2 \right ) \right \rangle$.

The subgroup $\left \langle \left ( 2,2 \right ) \right \rangle$ would absorb any elements of the form $n\left ( 2,2 \right )$ by the property of cosets.

Why wouldn't the element (1,2)H be in the factor group?

At this point I would appreciate hints. My intuition tells me a bit of number theory would be in use here.

Thanks in advance.

Answer 1

The element $(1,1) + H$ has order $2$.

The element $(1,2) + H$ has order $\infty$.

Therefore, the group is not cyclic.