Difficult# Infinite Cyclic Groups

ABSALG-EE0NKG

Let $G$ be a group. The index of a normal subgroup $K$ in $G$ is the cardinality of $G/K$.

For $g\in G$, let $\langle g\rangle$ be the cyclic subgroup generated by $g$. If we have $G=\langle g\rangle$ for some $g\in G$, then $G$ is called a cyclic group. If the order of the group $\langle g\rangle$ is finite, we say it is finite cyclic, otherwise we say it is infinite cyclic.

Let ${\mathbb Z}$ denote the group of integers under addition, and $2{\mathbb Z}$ the subgroup of ${\mathbb Z}$ consisting of the even integers under addition.

**Which of the following is FALSE**?