Question

Show that the additive group in $\square$is not cyclic [Hint: Exercise 49.]

Short answer

Additive group $\square$is not cyclic.

Step-by-step solution

Definition of a cyclic group

A cyclic group is a group that can be generated by a single element. That single element is known as a generator of a group.

Proving that additive group □is not cyclic,

Suppose Q is cyclic, so there must be $x\in \square$, such that role="math" localid="1651314604310" .

Here, x is the generator of the additive group$\square$.

Now, to prove that $\square$is not cyclic, we have to find a value of n, which does not satisfy the above condition.

To do that, let us suppose $\frac{1}{2}x\in Q$.

So,

$\frac{1}{2}x=nx$

From this, we get $n=\frac{1}{2}$.

This is not possible because $n\in Z$.

Therefore, our assumption is wrong and we can conclude that additive group $\square$ is not cyclic.