Question

In Exercises 4-8, list (if possible) or describe the elements of the given cyclic subgroup.

7.in the multiplicative group of nonzero rational numbers.

Short answer

The elements in the cyclic subgroup of$\mathrm{\mathbb{Q}}*$ is the set $\left\{2^k,k\in \mathrm{\mathbb{Z}}\right\}$.

Step-by-step solution

Cyclic Subgroup

Every element x in a group G, form a subgroup of all integer powers is called a cyclic subgroup of x.

Elements in ℚ*

$\mathrm{\mathbb{Q}}*$be the group of nonzero rational numbers under multiplication

Elements in the cyclic subgroup 2 of ℚ*.

Here, the multiplicative identity is 1.

By the definition, the element $2\in \mathrm{\mathbb{Q}}*$, form a subgroup of all integer powers as which is the set$\left\{1,2,4,8,16,\dots \right\}\mathrm{U}\left\{\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$

.

Therefore, the elements in the cyclic subgroup of$\mathrm{\mathbb{Q}}*$ is the set $\left\{2^k,k\in \mathrm{\mathbb{Z}}\right\}$.