Dihedral Group:D10 - Groupprops - Subwiki

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Definition

This group is defined as the dihedral group of order ten. In other words, it is the semidirect product of the cyclic group of order five and a cyclic group of order two.

Definition by presentation

The dihedral group D 10 {\displaystyle D_{10}} , sometimes called D 5 {\displaystyle D_{5}} , also called the dihedral group of order ten or the dihedral group of degree five (since its natural action is on five elements) is defined by the following presentation, with e {\displaystyle e} denoting the identity element:

⟨ x , a ∣ a 5 = x 2 = e , x a x − 1 = a − 1 ⟩ {\displaystyle \langle x,a\mid a^{5}=x^{2}=e,xax^{-1}=a^{-1}\rangle }

Here, the element a {\displaystyle a} is termed the rotation or the generator of the cyclic piece and x {\displaystyle x} is termed the reflection.

Arithmetic functions

Group properties

GAP implementation

Group ID

This finite group has order 10 and has ID 1 among the groups of order 10 in GAP's SmallGroup library. For context, there are groups of order 10. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(10,1)

For instance, we can use the following assignment in GAP to create the group and name it G {\displaystyle G} :

gap> G := SmallGroup(10,1);

Conversely, to check whether a given group G {\displaystyle G} is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [10,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

This group can be defined using GAP's DihedralGroup function:

DihedralGroup(10)