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From Groupprops

This article is about a particular group, viz a group unique upto isomorphism

This particular group is the smallest (in terms of order): non-T-group

This particular group is the smallest (in terms of order): nilpotent non-Abelian group

This particular group is a finite group of order: 8

Contents 1 Definition 1.1 Definition by presentation 1.2 Geometric definition 1.3 Multiplication table 1.4 Other definitions 2 Elements 2.1 Upto conjugacy 2.2 Upto automorphism 3 Group properties 3.1 Nilpotence 3.2 Solvability 3.3 Abelianness 3.4 Simplicity 3.5 ACIC 4 Subgroup-defining functions 4.1 Center 4.2 Commutator subgroup 4.3 Frattini subgroup 4.4 Socle 5 Subgroups 5.1 Normal subgroups 5.2 Characteristic subgroups 5.3 Conjugate-permutable subgroups 5.4 Permutable subgroups 5.5 Automorph-permutable subgroups 5.6 Automorph-conjugate subgroups 6 In larger groups 6.1 Occurrence as a subgroup 6.2 Occurrence as a quotient 7 GAP implementation 7.1 Group ID 7.2 Other descriptions

Definition

Definition by presentation

The dihedral group D₈, sometimes called D₄, also called the dihedral group of order eight or the dihedral group acting on four elements, is defined by the following presentation:

Geometric definition

The dihedral group D₈ (also called D₄) is defined as the group of all symmetries of the square (the regular 4-gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by a) and has four reflections each being an involution: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

Multiplication table

Here, e denotes the identity element, a is an element of order 4, and x is an element of order two that isn't equal to a², as in the above presentation.

Element	e	a	a2	a3	x	ax	a2x	a3x
e	e	a	a2	a3	x	ax	a2x	a3x
a	a	a2	a3	e	ax	a2x	a3x	x
a2	a2	a3	e	a	a2x	a3x	x	ax
a3	a3	e	a	a2	a3x	x	ax	a2x
x	x	a3x	a2x	ax	e	a3	a2	a
ax	ax	x	a3x	a2x	a	e	a3	a2
a2x	a2x	ax	x	a3x	a2	a	e	a3
a3x	a3x	a2x	ax	x	a3	a2	a	e

Other definitions

The dihedral group can be described in the following ways:

1. The dihedral group of order eight.
2. The holomorph of the cyclic group of order four.
3. The external wreath product of the cyclic group of order two with the cyclic group of order two, acting via the regular action.
4. The 2-Sylow subgroup of the symmetric group on four letters.
5. The 2-Sylow subgroup of the symmetric group on five letters.
6. The 2-Sylow subgroup of the alternating group on six letters.

Elements

Upto conjugacy

There are five conjugacy classes of elements of the dihedral group:

1. The identity element
2. The rotation by π, which is given as a² in the presentation
3. The two-element conjugacy class comprising rotations by , namely a and a³
4. The two-element conjugacy class comprising the two reflections: x,xa²
5. The two-element conjugacy class comprising the two reflections: xa,xa³

Upto automorphism

Under the equivalence relation of automorphisms, the last two conjugacy classes merge into one. There are thus four equivalence classes under the actions of automorphisms, of sizes 1, 1, 2 and 4.

Group properties

Nilpotence

This particular group is nilpotent

In fact, the upper central series and lower central series coincide, both of them being of length two, with the center and commutator subgroup both being the subgroup generated by a².

Solvability

This particular group is solvable

The group has solvable length 2. The derived series is the same as the upper central series and the lower central series.

Abelianness

This particular group is not Abelian

The group is not Abelian: .

Simplicity

This particular group is not simple

The group is not simple: it has normal subgroups of order 2 and 4.

ACIC

This group is an ACIC-group

The dihedral group of order eight is ACIC: any automorph-conjugate subgroup of it is characteristic. (The description of subgroups given below makes this clear).

Subgroup-defining functions

Center

The center of this group is abstractly isomorphic to: cyclic group of order two

The center of the quaternion group is the two-element subgroup comprising the identity and a² (rotation by π)

Commutator subgroup

The commutator subgroup of this group is abstractly isomorphic to: cyclic group of order two

The commutator subgroup of the dihedral group is the same as its center.

In particular this shows that the dihedral group is a group of nilpotence class two.

Frattini subgroup

The Frattini subgroup of this group is abstractly isomorphic to: cyclic group of order two

The Frattini subgroup coincides with the center and commutator subgroup. This dihedral group is thus an extraspecial group.

Socle

The socle of this group is abstractly isomorphic to: cyclic group of order two

The center is the unique minimal normal subgroup, and hence is also the socle.

Subgroups

Further information: subgroup structure of dihedral group:D8 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 a². (1)
3. The two-element subgroups generated by x, ax, a²x and a³x. (4)
4. The four-element subgroup generated by a² and x. This comprises elements e,a²,x,a²x. It is isomorphic to the Klein-four group. A similar four-element subgroup is obtained as that generated by a² and ax. (2)
5. The four-element subgroup generated by a. (1)
6. The whole group. (1)

Normal subgroups

All subgroups except those in header (3) above, are normal. The subgroups in header (3), which are two-element subgroups generated by some aⁿx, are 2-subnormal, as each of these is contained in a Klein-four group. Of the subgroups in header (3), there are two conjugacy classes: one comprising the subgroups generated by x and by a²x, and the other comprising the subgroups generated by ax and by a³x. (These conjugacy classes are related by an outer automorphism).

Characteristic subgroups

The subgroups in headers (1), (2), (5) and (6) are characteristic. The subgroups in header (4) are normal but not characteristic, and in fact, the two subgroups are automorphs of each other.

Conjugate-permutable subgroups

Every subgroup of the dihedral group of order eight is conjugate-permutable. The subgroups in all headers except (3) are anyway normal, while for the subgroups in header (3), any such subgroup commutes element-wise with its other conjugate subgroup.

Permutable subgroups

The permutable subgroups of this group are same as the normal subgroups. In other words, this is a N=P-group

The only permutable subgroups of the dihedral group are its normal subgroups. The subgroups in header (3) fail to permute with each other.

Automorph-permutable subgroups

In fact, the automorph-permutable subgroups are the same as the normal subgroups. That's because the subgroups in header (3) are automorphs of each other, but fail to commute with each other.

Automorph-conjugate subgroups

The automorph-conjugate subgroups are precisely the same as the characteristic subgroups. For this, observe that the subgroups in header (3) and (4) are not automorph-conjugate: some of them are automorphs but are not conjugate subgroups.

In larger groups

Occurrence as a subgroup

The dihedral group of order eight occurs as a subgroup in bigger groups. Here are some examples:

1. It is a subgroup in a dihedral group of order 2n where n is a multiple of 4.
2. It occurs as a Sylow subgroup in a number of groups: for instance, in the symmetric group on four letters.

Occurrence as a quotient

The dihedral group of order eight also occurs as a quotient; for instance, it is a quotient of the dicyclic group of order 16, by its center (which has order two).

GAP implementation

Group ID

The dihedral group of order eight has group ID 3. In other words, it can be described using the [[GAP:Smallas:

SmallGroup(8,3)

Other descriptions

The dihedral group can be described using GAP's DihedralGroup command:

DihedralGroup(8)

It can also be described as a wreath product:

WreathProduct(CyclicGroup(2),SymmetricGroup(2))

It can be described as the holomorph of the cyclic group of order four. For this, first define G to be the cyclic group of order four, and then use SemidirectProduct. Alternatively, use GAP:HolomorphOfGroup, a short piece of code to define a new function to compute the holomorph of a group.