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Dihedral group:D8

(Redirected from Dihedral group:D4)

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

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:

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

Geometric definition

The dihedral group 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.

Elements

Upto conjugation

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

• The identity element
• The rotation by π, which is given as a2 in the presentation
• The two-element conjugacy class comprising rotations by $\pm \pi/2$, namely a and a3
• The two-element conjugacy class comprising the two reflections: x,xa2
• 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.

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.