Subgroup structure of dihedral groups

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: dihedral group.
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We study here the subgroup structure of finite dihedral groups. See subgroup structure of infinite dihedral group for the subgroup structure of the infinite dihedral group.

For any natural number n, we define:

D_{2n} := \langle a,x \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle.

For n \ge 3, D_{2n} is the group of symmetries in the regular n-gon in the plane.

There are two kinds of subgroups:

  1. Subgroups of the form \langle a^d \rangle, where d | n. There is one such subgroup for each d. The total number of such subgroups is \tau(n) or \sigma_0(n), i.e., the number of positive divisors of n.
  2. Subgroups of the form \langle a^d, a^rx\rangle where d | n and 0 \le r < d. There are thus d such subgroups for each such divisor d. The total number of such subgroups is \sigma(n) or \sigma_1(n), i.e., the sum of positive divisors of n.

We consider various cases when discussing subgroup structure:

  • The special cases n = 1, n = 2, n = 4.
  • The special case where n is a power of 2.
  • The special case where n is odd.

The subgroup \langle a \rangle

Further information: Cyclic subgroup of dihedral group

The special case n = 1

In the case n = 1, the subgroup \langle a \rangle is trivial, and the whole group is cyclic of order two generated by x.

The special case of n = 2

In the case n = 2, the group D_{2n} is the Klein four-group:

D_4 = \langle a,x \mid a^2 = x^2 = e, ax = xa \rangle.

For this, the subgroup \langle a \rangle is a normal subgroup, but not a characteristic subgroup.

The case n \ge 3

In the case n \ge 3, the subgroup \langle a \rangle is the unique cyclic subgroup of order n. It satisfies the following properties:

These facts have a number of generalizations:

The case n \ne 1,2, 4

In the case n \ne 2, 4, the subgroup \langle a \rangle is the centralizer of commutator subgroup, i.e., it is the centralizer in D_{2n} of the commutator subgroup of D_{2n}, which is \langle a^2 \rangle.

There are a number of generalizations/related facts:

Odd versus even n

When n is odd, the cyclic subgroup of order n in the dihedral group of order 2n, satisfies the following properties:

None of these properties are satisfied when n is even.

Also, when n is odd, the cyclic subgroup of order n is the commutator subgroup.

The subgroup \langle x \rangle

General facts

Here are some general facts about this subgroup.

  • Its normalizer is \langle a^{n/2}, x \rangle when n is even. When n is odd, \langle x \rangle is self-normalizing.
  • The conjugate subgroups to this subgroup are subgroups of the form \langle a^{2r}x \rangle. When n is odd, this includes all subgroups of the form \langle a^k x \rangle, whereas if n is even, this includes only half the subgroups of the form \langle a^k x \rangle. There are thus n/2 conjugate subgroups if n is even and n conjugate subgroups if n is odd.
  • If n = 2 ,there are three automorphic subgroups, \langle a \rangle, \langle x \rangle, \langle ax \rangle. If n \ne 2, the automorphic subgroups to this subgroup are subgroups of the form \langle a^r x \rangle. There are thus n of them if n \ne 2.
  • The normal closure of \langle x \rangle is \langle a^2, x \rangle. This is the whole group if n is odd and is a subgroup of index two if n is even.

The special case of n = 2

When n = 2, the dihedral group is the Klein four-group, and \langle x \rangle is a normal subgroup. There is no other n for which \langle x \rangle is a normal subgroup.

The case n \ge 3

Some properties satisfied whenever n \ge 3 are:

Odd versus even n

When n is odd, then the subgroup \langle x \rangle satisfies the following properties:

On the other hand, when n is even, this subgroup satisfies none of these properties.

The case where n is a power of two

When n is a power of two, the subgroup \langle x \rangle satisfies the following properties: