Question

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

8.in the multiplicative group of nonzero elements of ${\mathrm{\mathbb{Z}}}_{11}$.

Short answer

The elements in the cyclic subgroup of ${\mathrm{\mathbb{Z}}}_{11}$is $\left\{1,3,4,5,9\right\}$.

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 ℤ11

The set of nonzero elements in${\mathrm{\mathbb{Z}}}_{11}$ is role="math" localid="1653466175066" ${\mathrm{\mathbb{Z}}}_{11}=\left\{1,2,3,4,5,6,7,8,9,10\right\}$.

Elements in the cyclic subgroup 3 of ℤ11.

For element $3\in {\mathrm{\mathbb{Z}}}_{11},3^1=3,3^2=9$,$3^3=27\equiv 5\left(mod11\right)$

$3^4=81=4\left(mod11\right),3^5=243\equiv 1\left(mod11\right)$

Thus, the cyclic subgroup generated by element 3 are .

Therefore, the elements in the cyclic subgroup of ${\mathrm{\mathbb{Z}}}_{11}$is $\left\{1,3,4,5,9\right\}$.