Question

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

6. 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 $\left\{1,2,3,4,5,6,7,8,9,10\right\}$.

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 ℤ11 

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

Elements in the cyclic subgroup 2 of ℤ11.

For element$$\begin{gathered} 2\in {\mathrm{\mathbb{Z}}}_{11},2^1=2,2^2=4,2^3=8,2^4=16\equiv 5\left(mod11\right), \\ {2}^5=32\equiv 10\left(mod11\right),2^6=64\equiv 9\left(mod11\right),2^7=128\equiv 7\left(mod11\right),2^8\equiv 3\left(mod11\right) \\ {2}^9\equiv 6\left(mod11\right),2^{10}\equiv 1\left(mod11\right). \end{gathered}$$

.

Thus, the cyclic subgroup generated by element 2 are which is the entire multiplicative group of nonzero elements of ${\mathrm{\mathbb{Z}}}_{11}$.

Therefore, the elements in the cyclic subgroup of${\mathrm{\mathbb{Z}}}_{11}$ is $\left\{1,2,3,4,5,6,7,8,9,10\right\}$.