Question

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

5. in the additive group $\mathrm{\mathbb{Z}}$.

Short answer

The elements in the cyclic subgroup of$\mathrm{\mathbb{Z}}$ is the set of even integers.

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 ℤ

The set of elements in $\mathrm{\mathbb{Z}}$is all the elements from $-\infty$to $+\infty$.

Elements in the cyclic subgroup 2 of ℤ.

Here, the additive identity is 0.

The, the cyclic subgroup generated by element 2 consists of all multiples of 2 so it is the set of even integers.

Therefore, the elements in the cyclic subgroup of$\mathrm{\mathbb{Z}}$ is the set of even integers.