2. find the order of the given group and the order of each element of the group. a. b.

Question

2. find the order of the given group and the order of each element of the group. a. < z₈, + mod 8 > b. < u(12), · mod 12 >

Accepted Answer

# Explanation:
## Step1: Recall group - order definition
The order of a finite group $G$, denoted as $|G|$, is the number of elements in the group.
## Step2: Find order of $\mathbb{Z}_8$
The group $\mathbb{Z}_8=\{0,1,2,3,4,5,6,7\}$ under addition modulo 8. So $|\mathbb{Z}_8| = 8$.
To find the order of an element $a\in\mathbb{Z}_8$, we find the smallest positive integer $n$ such that $na\equiv0\pmod{8}$.
 - For $a = 0$: $n = 1$ since $1\times0\equiv0\pmod{8}$, so the order of $0$ is $1$.
 - For $a = 1$: We need to find $n$ such that $n\times1\equiv0\pmod{8}$. The smallest positive $n$ is $8$, so the order of $1$ is $8$.
 - For $a = 2$: We solve $2n\equiv0\pmod{8}$, or $8\mid2n$. The smallest positive $n$ is $4$ since $2\times4 = 8\equiv0\pmod{8}$, so the order of $2$ is $4$.
 - For $a = 3$: We solve $3n\equiv0\pmod{8}$. We check $n = 1,2,\cdots$. Since $\gcd(3,8)=1$, the smallest positive $n$ such that $3n\equiv0\pmod{8}$ is $8$, so the order of $3$ is $8$.
 - For $a = 4$: We solve $4n\equiv0\pmod{8}$, or $8\mid4n$. The smallest positive $n$ is $2$ since $4\times2=8\equiv0\pmod{8}$, so the order of $4$ is $2$.
 - For $a = 5$: Since $\gcd(5,8) = 1$, the smallest positive $n$ such that $5n\equiv0\pmod{8}$ is $8$, so the order of $5$ is $8$.
 - For $a = 6$: We solve $6n\equiv0\pmod{8}$, or $3n\equiv0\pmod{4}$. The smallest positive $n$ is $4$ since $6\times4=24\equiv0\pmod{8}$, so the order of $6$ is $4$.
 - For $a = 7$: Since $\gcd(7,8)=1$, the smallest positive $n$ such that $7n\equiv0\pmod{8}$ is $8$, so the order of $7$ is $8$.
## Step3: Recall the definition of $U(12)$
The group $U(12)$ is the group of units modulo $12$, and $U(12)=\{1,5,7,11\}$ under multiplication modulo $12$. So $|U(12)| = 4$.
To find the order of an element $a\in U(12)$, we find the smallest positive integer $n$ such that $a^n\equiv1\pmod{12}$.
 - For $a = 1$: $1^1\equiv1\pmod{12}$, so the order of $1$ is $1$.
 - For $a = 5$: $5^1\equiv5\pmod{12}$, $5^2 = 25\equiv1\pmod{12}$, so the order of $5$ is $2$.
 - For $a = 7$: $7^1\equiv7\pmod{12}$, $7^2=49\equiv1\pmod{12}$, so the order of $7$ is $2$.
 - For $a = 11$: $11^1\equiv11\pmod{12}$, $11^2=( - 1)^2\equiv1\pmod{12}$, so the order of $11$ is $2$.

# Answer:
a. Order of $\langle\mathbb{Z}_8,+\bmod{8}
angle$ is $8$.
Order of $0$ is $1$, order of $1$ is $8$, order of $2$ is $4$, order of $3$ is $8$, order of $4$ is $2$, order of $5$ is $8$, order of $6$ is $4$, order of $7$ is $8$.
b. Order of $\langle U(12),\cdot\bmod{12}
angle$ is $4$.
Order of $1$ is $1$, order of $5$ is $2$, order of $7$ is $2$, order of $11$ is $2$.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall group - order definition

Step2: Find order of \(\mathbb{Z}_8\)

Step3: Recall the definition of \(U(12)\)

Answer: