# Subgroup structure of dihedral groups

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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

• 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:

• Subnormal subgroup: In particular, when $n = 2^k$, $\langle x \rangle$ is a $k$-subnormal subgroup.