Center of dihedral group:D8

This article discuss the dihedral group of order eight and its center, which is a cyclic group of order two.

The dihedral group of order eight is defined as:

$$G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$$.

It has multiplication table:

and the center is the cyclic subgroup:

$$H := \{ a^2, e \} = \langle a^2 \rangle$$.

It has multiplication table:



Cosets
The subgroup has the following four cosets:

$$\! \{ e, a^2 \}, \qquad \{ x, a^2x \}, \qquad \{ a, a^3 \}, \qquad \{ ax, a^3x \}$$

The quotient group is isomorphic to Klein four-group, and the multiplication table is as follows:

Complements
The subgroup $$H$$ has no permutable complement and also has no lattice complement.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.

Cohomology interpretation
We can think of $$G$$ as an extension with abelian normal subgroup $$H$$ and quotient group $$G/H$$. Since $$H$$ is in fact the center, the action of the quotient group on the normal subgroup is the trivial group action. We can thus study $$G$$ as an extension group arising from a cohomology class for the trivial group action of $$G/H$$ (which is a Klein four-group) on $$H$$ (which is cyclic group:Z2).

For more, see second cohomology group for trivial group action of V4 on Z2.

GAP implementation
The group and subgroup pair can be defined using GAP's DihedralGroup and Center functions as follows:

G := DihedralGroup(8); H := Center(G);

The GAP display looks as follows:

gap> G := DihedralGroup(8); H := Center(G);  Group([ f3 ])