- #36
Oxymoron
- 870
- 0
Ok I may have something.
Dihedral groups are special groups that consist of rotations [tex]a[/tex] and relfections [tex]b[/tex], where the group operation is the composition of these rotations and reflections.
The finite dihedral group [tex]\mathcal{D}_n[/tex] has [tex]2n[/tex] elements and is generated by [tex]a[/tex] (with order [tex]n[/tex]) and [tex]b[/tex] (with order [tex]2[/tex]). The two elements of the dihedral group satsify
[tex]ab = ba^{-1}[/tex]
If the order of [tex]\mathcal{D}_{n}[/tex] is greater than 4, then the group operations do not commute, ie [tex]\mathcal{D}_n[/tex] is not abelian.
The [tex]2n[/tex] elements of [tex]\mathcal{D}_n[/tex] are
[tex]\{e, a, a^2, \dots , a^{n-1}, b, ba, ba^2, \dots , ba^{n-1}\}[/tex]
Now, in order to form the quotient groups, I need to find the normal subgroups. The normal subgroups are those which are invariant under conjugation.
I know [tex]\{e\}[/tex] and [tex]\{\mathcal{D}_n\][/tex] are going to normal subgroups. But I don't know how to find any others.
Dihedral groups are special groups that consist of rotations [tex]a[/tex] and relfections [tex]b[/tex], where the group operation is the composition of these rotations and reflections.
The finite dihedral group [tex]\mathcal{D}_n[/tex] has [tex]2n[/tex] elements and is generated by [tex]a[/tex] (with order [tex]n[/tex]) and [tex]b[/tex] (with order [tex]2[/tex]). The two elements of the dihedral group satsify
[tex]ab = ba^{-1}[/tex]
If the order of [tex]\mathcal{D}_{n}[/tex] is greater than 4, then the group operations do not commute, ie [tex]\mathcal{D}_n[/tex] is not abelian.
The [tex]2n[/tex] elements of [tex]\mathcal{D}_n[/tex] are
[tex]\{e, a, a^2, \dots , a^{n-1}, b, ba, ba^2, \dots , ba^{n-1}\}[/tex]
Now, in order to form the quotient groups, I need to find the normal subgroups. The normal subgroups are those which are invariant under conjugation.
I know [tex]\{e\}[/tex] and [tex]\{\mathcal{D}_n\][/tex] are going to normal subgroups. But I don't know how to find any others.