Orders of Elements in Groups $$\Bbb{Z}_{12}, U(10), U(12), D4$$

In summary, the conversation discusses finding the order of different groups and elements, specifically in $\mathbb{Z}_{12}$. The order of the group is 12, and the order of each element is also listed. The concept of order in a group is explained using an example.
  • #1
karush
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For each group in the following list,
$$ \Bbb{Z}_{12}, \qquad U(10)\qquad U(12) \qquad D4 $$
(a) find the order of the group
$$|\Bbb{Z}_{12}|=12$$
(b) the order of each element in the group.ok the eq I think we are supposed to use is
$$\textit{ if } o(g)=n \textit{ then } o(g^n)= n/(n,k)$$
the alleged a answer for (a) is $\Bbb{Z}_{12}$
for (b)$o(0)=1, \quad $o(1)=12$ \quad $o(2)=6$\quad $o(3)=4$\quad $o(4)=3$,\quad $o(5)=12$
\quad $o(6)=2$\quad $o(7)=12$\quad $o(8)=3$\quad $o(9)=4$\quad $o(10)=6$\quad $o(11)=12$I am sure this is simple but don't see it
 
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  • #2
Hi karush,

The order an element in $\mathbb Z_{12}$is the number of times we need to add the element before we get 0 modulo 12.
For instance $(4+4+4) \bmod{12}=12 \bmod{12}=0$.
So the order of $4$ is $o(4)=3$.
 
  • #3
I like Serena said:
Hi karush,
The order an element in $\mathbb Z_{12}$is the number of times we need to add the element before we get 0 modulo 12.
For instance $(4+4+4) \bmod{12}=12 \bmod{12}=0$.
So the order of $4$ is $o(4)=3$.
so then
$o(1)=12$ is $(1+1+1+1 +1+1+1+1 +1+1+1+1) \bmod{12}=0$
and
$o(2)=6$ is $(2+2+2+2+2+2+2) \bmod{12} =0$
 

Related to Orders of Elements in Groups $$\Bbb{Z}_{12}, U(10), U(12), D4$$

1. What are the orders of elements in the group $\Bbb{Z}_{12}$?

The orders of elements in the group $\Bbb{Z}_{12}$ are 1, 2, 3, 4, 6, and 12. This means that the elements in this group can be raised to these powers and result in the identity element, which is 0 in this case.

2. What is the order of the group $U(10)$?

The order of the group $U(10)$, also known as the group of units modulo 10, is 4. This means that there are 4 elements in this group that are relatively prime to 10.

3. What are the orders of elements in the group $U(12)$?

The orders of elements in the group $U(12)$ are 1, 2, 3, 4, 6, and 12. This is because 12 is a relatively prime number, so all elements in this group are relatively prime to 12 and can be raised to these powers to result in the identity element.

4. What is the order of the group $D4$?

The order of the group $D4$, also known as the dihedral group of order 8, is 8. This means that there are 8 elements in this group, which are the rotations and reflections of a square.

5. What is the significance of the orders of elements in a group?

The order of an element in a group represents the smallest power at which that element becomes the identity element. This is important in understanding the structure and properties of a group, such as its subgroups and cyclic nature. It also helps in determining the number of elements in a group and their relationships with each other.

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