Chapter 4, Problem 1E

Find all generators of $Z_6,Z_8,\text{ and}{Z}_{20}$ .

To determine

To find:The generators of $Z_6,Z_{8}and{Z}_{20}$ .

Answer to Problem 1E

The generator of $Z_6$ is $\left\{1,5\right\}$ , generator of $Z_8$ is $\left\{1,3,5,7\right\}$ and $Z_{20}$ is $\left\{1,3,7,9,11,13,17,19\right\}$

Explanation of Solution

Given information: $\left(Z_n,{\oplus }_n\right)$ is a group under addition modulo n.

For a group G, if there exist an element $a\in G$ such that $G=\left\{a^n:n\in Z\right\}$ , then such an element a is called generator of the group.

In the group $\left(Z_n,{\oplus }_n\right),$ an integer k is said to be generator of $Z_n$ if and only if $\gcd \left(k,n\right)=1$ .

For group $Z_6,$ the set of integers whose $\gcd \left(k,n\right)=1$ is $\left\{1,5\right\}$

Therefore, in $Z_6,$ there are two generators given by $\left\{1,5\right\}$ .

Similarly,

For group $Z_8,$ the set of integers whose $\gcd \left(k,n\right)=1$ is $\left\{1,3,5,7\right\}$

Therefore, in $Z_6,$ there are four generators given by $\left\{1,3,5,7\right\}$ .

And

For group $Z_{20},$ the set of integers whose $\gcd \left(k,n\right)=1$ is $\left\{1,3,7,9,11,13,17,19\right\}$

Therefore, in $Z_6,$ there are eight generators given by $\left\{1,3,7,9,11,13,17,19\right\}$ .