Group theory and generalizations Example

Orders of the elements of the integers modulo 6

Builds on Integers modulo n Order of an element

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 $6$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

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

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

Orders of the elements of the integers modulo 6

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here