Order of the dihedral group

Order of the dihedral group
Tag	`0BZVNWJ352`
URI	`m2://0BZVNWJ352`
Kind	statement, proposition
Writing	W0 stub
MSC	20B25
Status	This copy of the site has no build data. Run `m2 status 0BZVNWJ352`.
Uses	Dihedral group, Order of a group

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Statement

Let $n \geq 3$ . The dihedral group $D_n$ has order $2n$ .

Remarks

A symmetry of the regular $n$ -gon is determined by the image of one vertex and by whether it keeps or reverses the orientation. There are $n$ choices for the image of the vertex and $2$ choices for the orientation. For example, $D_3$ has order $6$ and $D_4$ has order $8$ .

Explanations

L1 human

A symmetry can send a fixed vertex to any of the $n$ vertices. After that, the polygon can be face up or face down. So there are $n \cdot 2 = 2n$ symmetries.

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Group concept
Symmetry group concept
Order of a group definition
Subgroup definition
Subgroup criterion statement
Intersection of subgroups is a subgroup statement
Subgroup generated by a subset definition
Dihedral group definition
Order of the dihedral group statement

Sources

Web https://en.wikipedia.org/wiki/Dihedral_group, Definition

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