Klein four-subgroups of dihedral group:D8

This article discusses the dihedral group of order eight (see details on the subgroup structure) and the two Klein four-subgroups of this group. We call the dihedral group $$G$$, and use the following presentation:

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

and we set:

$$\! H_1 = \langle a^2,x \rangle, \qquad H_2 = \langle a^2,ax \rangle $$.

As full lists:

$$\! H_1 = \{ e, x, a^2, a^2x \}, \qquad H_2 = \{ e, ax, a^2, a^3x \}$$



Cosets
Both subgroups have index two and are hence normal subgroups. Each of them has two cosets: the subgroup itself and the rest of the group.

The cosets of $$H_1$$ are:

$$\! H_1 = \{ e, x, a^2, a^2x \}, G \setminus H_1 = \{ ax, a^3x, a, a^3 \}$$

The cosets of $$H_2$$ are:

$$\! H_2 = \{ e, ax, a^2, a^3x \}, G \setminus H_1 = \{ x, a^2x, a, a^3 \}$$

Complements
$$H_1$$ has two permutable complements: $$\{ e,ax \}$$ and $$\{ e, a^3x \}$$, both of which are conjugate subgroups and subgroups of $$H_2$$. $$H_2$$ has two permutable complements: $$\{ e, x \}$$ and $$\{ e, a^2x \}$$, both of which are conjugate subgroups and are subgroups of $$H_1$$.

The four subgroups of order two arising as complements in this way are all automorphic subgroups. For information on these, see non-normal subgroups of dihedral group:D8.

Properties related to complementation
For convenience, we refer to $$H_1$$ below, though $$H_2$$ behaves the same way.

Effect of subgroup operators
In the table below, we provide values specific to $$H_1$$.

Automorphism class-defining functions
Below are some functions that start with the whole group as a black box group and give the unique automorphism class comprising the two Klein four-subgroups. Note that these functions are not guaranteed to always give a single automorphism class of subgroups for a general group.

Intermediate subgroups
There are no properly intermediate subgroups because the subgroup is a maximal subgroup.

Smaller subgroups
Below we discuss subgroups inside $$H_1$$.

Images under quotient maps
The discussion below is for $$H_1 = \langle a^2, x \rangle$$.

Invariance under automorphisms and endomorphisms
Suppose $$c_a$$ and $$c_x$$ denote conjugation by $$a$$ and $$x$$ respectively. Let $$\sigma$$ denote the automorphism that sends $$a$$ to $$a^3$$ and $$x$$ to $$ax$$. Then, $$\langle c_a, c_x\rangle$$ is the inner automorphism group and $$\langle c_a, c_x, \sigma \rangle$$ is the automorphism group. It turns out that $$\sigma(H_1) = H_2$$ and $$\sigma(H_2) = H_1$$, while $$c_a$$ and $$c_x$$ preserve both $$H_1$$ and $$H_2$$. Note that the automorphism group is isomorphic to dihedral group:D8 and the inner automorphism group to the Klein four-group.

Note that since $$\sigma$$ sends $$H_1$$ to $$H_2$$, they both satisfy the same subgroup properties. For convenience, we use notation and symbols for $$H_1$$.

Generic maximality notions
$$H_1$$ and $$H_2$$ are both maximal subgroups of the group of prime power order $$G$$. Thus, they satisfy all these properties: satisfies property::maximal normal subgroup, satisfies property::maximal subgroup, satisfies property::subgroup of index two, satisfies property::order-normal subgroup, satisfies property::isomorph-normal subgroup, satisfies property::maximal subgroup of finite nilpotent group.

As already discussed, they are both also coprime automorphism-invariant, hence they are satisfies property::isomorph-normal coprime automorphism-invariant subgroup of group of prime power order. In particular, they are both satisfies property::fusion system-relatively weakly closed subgroups and thus satisfies property::Sylow-relatively weakly closed subgroups.

Abelian subgroups of maximum order
$$H_1$$ and $$H_2$$ are both abelian subgroups of maximum order. There is one more abelian subgroup of maximum order, namely the cyclic subgroup $$\langle a \rangle$$.

Elementary abelian subgroups of maximum order
$$H_1$$ and $$H_2$$ are both elementary abelian subgroups of maximum order in $$G$$. There are no others.

Abelian subgroups of maximum rank
$$H_1$$ and $$H_2$$ are both abelian subgroups of maximum rank in $$G$$. There are no others.

Finding these subgroups inside a black-box dihedral group
Suppose $$G$$ is a group that we know is isomorphic to dihedral group:D8. We can create a two-element list of the Klein four-subgroups of $$G$$ using the command:

H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]);

We can then set:

H1 := H[1]; H2 := H[2];

Constructing the dihedral group and the two subgroups
This can be achieved by the sequence of commands:

G := DihedralGroup(8); H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]); H1 := H[1]; H2 := H[2];