Subgroup structure of dihedral group:D8

The dihedral group $D_8$, sometimes called $D_4$, also called the dihedral group of order eight or the dihedral group acting on four elements, is defined by the following presentation:

$⟨x,a|a^4=x^2=1,xa{x}^{-1}=a^{-1}⟩$

The dihedral group has ten subgroups:

1. The trivial subgroup (1)
2. The center, which is the unique minimal normal subgroup, and is a two-element subgroup generated by $a^2$. (1)
3. The two-element subgroups generated by $x$, $ax$, $a^{2}x$ and $a^{3}x$. (4)
4. The four-element subgroup generated by $a^2$ and $x$. This comprises elements $e,a^2,x,a^{2}x$. It is isomorphic to the Klein-four group. A similar four-element subgroup is obtained as that generated by $a^2$ and $ax$. (2)
5. The four-element subgroup generated by $a$. (1)
6. The whole group. (1)

We study here the properties of each of these subgroups (except the trivial subgroup and the whole group). We denote the whole group by $G$.

First, a quick summary:

• Except the subgroups in (3), all subgroups are normal. Of the subgroups listed in (3), there are two conjugacy classes of subgroups, each comprising two subgroups. Both conjugacy classes are related by an outer automorphism.
• The subgroups listed in (1), (2), (5) and (6) are characteristic. The two subgroups listed in (4) are normal, but are automorphs of each other.

The center (type (2))

This is a two-element subgroup $\{a^2,e\}$. It is a characteristic subgroup.

Subgroup-defining functions yielding this subgroup

There are many subgroup-defining functions that yield this subgroup, for instance:

• The center: $Z(G)=\{a^2,e\}$. These are the only two elements that commute with every element. It is also equal to ${\mathrm{\Omega }}^1(Z(G))$, the subgroup generated by elements of order two in the center.
• The commutator subgroup: $[G,G]=\{a^2,e\}$. The quotient group is isomorphic to the Klein-four group.
• The Frattini subgroup: It is the intersection of three maximal subgroups, each of order four. (these are covered in points (4) and (5) in the list).
• The socle: In fact, $\{a^2,e\}$ is the unique minimal normal subgroup.
• The first agemo subgroup: $\{a^2,e\}$ is the subgroup generated by all squares, and is hence $\stackrel{\mathrm{g}}{\mathrm{A}}(G)$.

Subgroup properties satisfied by this subgroup

On account of being an agemo subgroup as well as on account of being the commutator subgroup, the center is a verbal subgroup -- it is a subgroup generated by words of a certain form (in the agemo description, these words are squares; in the commutator subgroup description, these words are commutators). Thus, it satisfies the following properties:

• Fully characteristic subgroup: It is invariant under any endomorphism of the whole group. Further information: Verbal implies fully characteristic
• Image-closed fully characteristic subgroup: Its image under any surjective homomorphism is fully characteristic in the image. Further information: Verbal implies image-closed fully characteristic
• Image-closed characteristic subgroup: Its image under any surjective homomorphism is characteristic in the image. Further information: Verbal implies image-closed characteristic
• Characteristic subgroup: It is characteristic in the whole group.

Subgroup properties not satisfied by this subgroup

• Intermediately characteristic subgroup: The subgroup $\{a^2,e\}$ is not characteristic in every intermediate subgroup. In particular, it is not characteristic in subgroups of the type (4), such as $\{e,a^2,x,a^{2}x\}$. Further information: Characteristicity does not satisfy intermediate subgroup condition, center not is intermediately characteristic, commutator subgroup not is intermediately characteristic, Frattini subgroup not is intermediately characteristic

The four-element characteristic subgroup (type (5))

Subgroup-defining functions yielding this subgroup

None of the standard choices of subgroup-defining functions yields this subgroup. It can be described using the following:

• It is the unique maximal among Abelian characteristic subgroups. Note that for a general group, there need not exist a unique maximal among Abelian characteristic subgroups.
• It is the unique constructibly critical subgroup, i.e., the only subgroup that could arise through an application of the constructive procedure in Thompson's critical subgroup theorem. However, it is not the only critical subgroup.

Subgroup properties satisfied by this subgroup

The subgroup is a cyclic maximal subgroup. On account of this, it satisfies the following:

• Intermediately characteristic subgroup: This subgroup is characteristic in every intermediate subgroup.

Subgroup properties not satisfied by this subgroup

• Fully characteristic subgroup, retraction-invariant subgroup: There exists a retraction with kernel $\{e,a^2,x,a^{2}x\}$ and image $\{e,ax\}$ that does not leave this subgroup invariant. Hence, it is not fully characteristic or retraction-invariant.
• Verbal subgroup: Since the subgroup is not fully characteristic, it is not verbal.
• Image-closed characteristic subgroup: Quotienting out by the center of the whole group gives a subgroup that is not characteristic in the image. Further information: Characteristicity does not satisfy image condition