Question

Show that the multiplicative group $\square *$of nonzero real numbers is not cyclic.

Short answer

The multiplicative group$\square *$ of non-zero real numbers is not cyclic.

Step-by-step solution

Definition of cyclic group

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

Prove that the multiplicative group □*is not cyclic

Suppose the multiplicative group$\square *$ is cyclic.

So, there must be some x such that,

Since .$-1\in \square$So, it should satisfy the condition,

${\mathrm{x}}^{-1}={\mathrm{x}}^{\mathrm{n}}$

Which implies that, n = -1 .

Since ncannot be -1 therefore, our assumption is wrong.

Hence, the multiplicative group$\square *$ of non-zero real numbers is not cyclic.