Question

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

4.in the additive group ${\mathrm{\mathbb{Z}}}_{12}$.

Short answer

The elements in the cyclic subgroup of${\mathrm{\mathbb{Z}}}_{12}$ is $\left\{0,2,4,6,8,10\right\}$.

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 ℤ12

The set of elements in${\mathrm{\mathbb{Z}}}_{12}$ is role="math" localid="1653463328630" ${\mathrm{\mathbb{Z}}}_{12}=\left\{0,1,2,3,4,5,6,7,8,9,10,11\right\}$.

Elements in the cyclic subgroup 2 of ℤ12.

Here, the additive identity is 0.

For element $2\in {\mathrm{\mathbb{Z}}}_{12}$, 2 + 0 = 2 , 2 + 2 = 4 , 4 + 2 = 6 , 6 +2 = 8 , 8 + 2 = 10 , $10+2=12\equiv 0\left(mod12\right)$.

.

Thus, the cyclic subgroup generated by element 2 are which consists of all multiples of 2.

Therefore, the elements in the cyclic subgroup of${\mathrm{\mathbb{Z}}}_{12}$ is $\left\{0,2,4,6,8,10\right\}$.