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

• MHB
• karush
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.
karush
Gold Member
MHB
nmh{909}
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 $\Bbb{Z}_{12}$

## 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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