Orders of the elements of the integers modulo 6

Orders of the elements of the integers modulo 6
Tag	`77N4TVRM15`
URI	`m2://77N4TVRM15`
Kind	example
Writing	W0 stub
MSC	20A05
Instance of	Order of an element
Illustrates	Order of an element divides the order of the group
Uses	Integers modulo n, Order of an element

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Example

Let $\mathbb{Z}/6\mathbb{Z} = \{0, 1, 2, 3, 4, 5\}$ be the integers modulo $6$ with addition. The element $2$ has order $3$ , because $2 + 2 + 2 = 6 = 0$ and $2 + 2 = 4 \ne 0$ . The element $3$ has order $2$ , and the elements $1$ and $5$ have order $6$ . The element $4$ has order $3$ . Each of the orders $1$ , $2$ , $3$ and $6$ divides $6$ , as the result on element orders states.

Remarks

The group is abelian, but not each element has order $6$ .

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Group concept
Integers modulo n concept
Laws of exponents in a group statement
Order of an element definition
Orders of the elements of the integers modulo 6 example

Sources

Web https://en.wikipedia.org/wiki/Order_(group_theory), Order and structure

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