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Abstract Algebra

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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?

A

Every infinite cyclic group is isomorphic to $2{\mathbb Z}$.

B

Every non-trivial subgroup of an infinite cyclic group is isomorphic to $2{\mathbb Z}$.

C

For every integer $m\ge 1$, there is a unique subgroup of $2{\mathbb Z}$ that is of index $m$ in ${\mathbb Z}$.

D

Every finite cyclic group is isomorphic to the quotient of $2{\mathbb Z}$ by a normal subgroup of ${\mathbb Z}$.

E

Every infinite cyclic group is isomorphic to ${\mathbb Z}$.