TITLE: In abelian groups: Is $[G:2G]\leq 2$?
QUESTION [5 upvotes]: Let $G$ be an abelian group. Is it true that $[G:2G]\leq 2$?
Sometimes it can be $1$ (say, when $G=\mathbb{Z}_3$), but it seems never to be more than $2$.

REPLY [4 votes]: Hint: Given any two abelian groups $G$ and $H$, if we let $K=G\times H$, then $2K=2G\times 2H$, and
$$K/2K\cong (G/2G)\times(H/2H).$$
What does this mean about the cardinality of $K/2K$, i.e., the index $[K:2K]$? Does that give you any ideas for finding a counterexample to your conjecture?