Subgroup structure of dihedral group:D8

From Groupprops
Revision as of 22:44, 20 August 2008 by Vipul (talk | contribs)
Jump to: navigation, search

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:

\langle x,a| a^4 = x^2 = 1, xax^{-1} = a^{-1}\rangle


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^2x and a^3x. (4)
  4. The four-element subgroup generated by a^2 and x. This comprises elements e,a^2,x,a^2x. 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 \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 \operatorname{Agemo}^1(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:

Subgroup properties not satisfied by this subgroup

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:

Subgroup properties satisfied by this subgroup

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

Subgroup properties not satisfied by this subgroup

The non-characteristic four-element subgroups (type (4))

These two subgroups are related by an outer automorphism, but are not conjugate (in fact, both are normal subgroups). Since they're automorphs, they in particular satisfy and dissatisfy the same subgroup properties.

Subgroup properties satisfied by these subgroups

  • Maximal normal subgroup: They are normal subgroups of index two.
  • Potentially characteristic subgroup: We can embed the dihedral group inside the symmetric group on four letters in such a way that one of these subgroups becomes characteristic (we cannot do this simultaneously for both). In fact, if we consider the usual action of the dihedral group on a square by symmetries, then one of these subgroups comprises the double transpositions, and this is characteristic in the group of all permutations on four letters.

Subgroup properties not satisfied by these subgroups

  • Automorph-conjugate subgroup: Although the two subgroups are automorphs of each other, they are not conjugate. Hence, neither of them is an automorph-conjugate subgroup.

The two-element non-normal subgroups (type (3))

Subgroup properties satisfied by these subgroups

Subgroup properties not satisfied by these subgroups