Dihedral group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dicyclic group (also called binary dihedral group)
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This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

Definition

The dihedral group of degree $n$ and order $2n$, denoted sometimes as sometimes as $D_{2n}$ (this wiki uses $D_{2n}$), sometimes as $D_n$, and sometimes as $\operatorname{Dih}_n$, is defined in the following equivalent ways:

• It has the presentation (here, $e$ denotes the identity element):

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

• (For $n \ge 3$): It is the group of symmetries of a regular $n$-gon in the plane, viz., the plane isometries that preserves the set of points of the regular $n$-gon.

The dihedral groups arise as a special case of a family of groups called von Dyck groups. They also arise as a special case of a family of groups called Coxeter groups.

Note that for $n = 1$ and $n = 2$, the geometric description of the dihedral group does not make sense. In these cases, we use the algebraic description.

The infinite dihedral group, which is the $n = \infty$ case of the dihedral group and is denoted $D_\infty$ and is defined as:

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

Particular cases

For small values

Note that all dihedral groups are metacyclic and hence supersolvable. A dihedral group is nilpotent if and only if it is of order $2^k$ for some $k$. It is abelian only if it has order $2$ or $4$.

Order of group Degree (size of regular polygon it acts on) Common name for the group Comment
2 1 Cyclic group:Z2 Not usually considered a dihedral group.
4 2 Klein four-group elementary abelian group that is not cyclic
6 3 symmetric group:S3 metacyclic, hence supersolvable but not nilpotent
8 4 dihedral group:D8 nilpotent but not abelian
10 5 dihedral group:D10 metacyclic, hence supersolvable but not nilpotent
12 6 dihedral group:D12 direct product of the dihedral group of order six and the cyclic group of order two.
16 8 dihedral group:D16 nilpotent but not abelian

Arithmetic functions

$n$ here denotes the degree, or half the order, of the dihedral group, which we denote as $D_{2n}$.

Function Value Explanation
order $2n$ Cyclic subgroup of order $n$, quotient of order $2$.
exponent $\operatorname{lcm}(n,2)$ Exponent of cyclic subgroup is $n$, elements outside have order $2$.
derived length $2$ for $n \ge 3$, $1$ for $n = 1,2$
nilpotency class $k$ when $n = 2^k$, none otherwise Nilpotent only if $n$ is a power of $2$.
max-length 1 + sum of exponents of prime divisors of $n$ The dihedral groups are solvable.
composition length 1 + sum of exponents of prime divisors of $n$ The dihedral groups are solvable.
number of subgroups $\sigma(n) + d(n)$ $\sigma(n)$ is the divisor sum function and $d(n)$ is the divisor count function.
number of conjugacy classes $(n+3)/2$ if $n$ is odd, $(n+6)/2$ if $n$ is even.
number of conjugacy classes of subgroups $3d(n) - d(m)$ where $m$ is the largest odd divisor of $n$ $d(n)$ subgroups inside the cyclic part. For odd divisors, one external conjugacy class of subgroups per divisor; for even divisors, one external conjugacy class per divisor.

Group properties

Property Satisfied? Explanation
Abelian group no for $n \ge 3$ For $n \ge 3$, the elements $a,x$ do not commute.
Nilpotent group yes for $n$ a power of $2$, no otherwise
Metacyclic group yes
Supersolvable group yes
Solvable group yes
T-group yes for $n$ odd or twice an odd number, no for $n$ a multiple of $4$
Rational group yes for $n \le 4$ and $n = 6$, no for $n = 5$, $n \ge 7$ The field generated by character values is $\mathbb{Q}(\cos(2\pi / n))$. The number $\cos(2\pi/n)$ is rational if and only if $n = 1, 2, 3, 4, 6$. This can be proved by showing that $2\cos(2\pi/n)$ is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in $\{ -2, -1, 0, 1, 2 \}$, which translates to these five values of $n$. For more information, see linear representation theory of dihedral groups.
Rational-representation group yes for $n \le 4$ and $n = 6$, no for $n = 5$, $n \ge 7$ The unique minimal splitting field is $\mathbb{Q}(\cos(2\pi / n))$. The number $\cos(2\pi/n)$ is rational if and only if $n = 1, 2, 3, 4, 6$. This can be proved by showing that $2\cos(2\pi/n)$ is an algebraic integer (it is the root of a monic polynomial with integer coefficients) and therefore must be in $\{ -2, -1, 0, 1, 2 \}$, which translates to these five values of $n$. For more information, see linear representation theory of dihedral groups.
Ambivalent group yes Elements in the cyclic subgroup are conjugate via $x$, elements outside have order two.

Elements

Further information: element structure of dihedral groups

Summary

Item Value
order $2n$
conjugacy class sizes Case $n$ odd: 1 (1 time), 2 ($(n-1)/2$ times), $n$ (1 time)
Case $n$ even: 1 (2 times), 2 ($(n-2)/2$ times), $n/2$ (2 times)
number of conjugacy classes $(n + 3)/2$ if $n$ odd, $(n + 6)/2$ if $n$ even
number of orbits under automorphism group $d(n) + 1$ where $d$ is the divisor count function if $n \ge 3$, 2 if $n = 2$
order statistics Case $n$ odd: $\varphi(d)$ of order $d$ for $d | n$, $n$ of order 2

Conjugacy class structure

Nature of conjugacy class Size of each conjugacy class Number of such conjugacy classes Total number of elements
Identity element 1 1 1
Non-identity elements in cyclic subgroup $\langle a \rangle$, where each element and its inverse form a conjugacy class 2 $(n - 1)/2$ $n - 1$
Elements outside the cyclic subgroup $\langle a \rangle$, all form a single conjugacy class $n$ 1 $n$
Total -- $(n + 3)/2$ (number of conjugacy classes) $2n$ (order of group)

Subgroups

Further information: Subgroup structure of dihedral groups

There are two kinds of subgroups:

• Subgroups of the form $\langle a^d \rangle$, where $d|n$. The number of such subgroups equals the number of positive divisors of $n$, sometimes denoted $\tau(n)$. The subgroup generated by $a^d$ is a cyclic group of order $n/d$.
• Subgroups of the form $\langle a^d, a^r x \rangle$, where $d|n$ and $0 \le r < d$. The number of such subgroups equals the sum of all positive divisors of $n$, sometimes denoted $\sigma(n)$. The subgroup of the above form is a dihedral group of order $2n/d$.

In particular, all subgroups of the dihedral group are either cyclic or dihedral.

Also note that the dihedral group has subgroups of all orders dividing its order. This is true more generally for all finite supersolvable groups. Further information: Finite supersolvable implies subgroups of all orders dividing the group order

Linear representation theory

Further information: linear representation theory of dihedral groups

Item Value
degrees of irreducible representations over a splitting field Case $n$ odd: 1 (2 times), 2 ($(n - 1)/2$ times)
Case $n$ even: 1 (4 times), 2 ($(n - 2)/2$ times)
maximum: 2 (if $n \ge 3$), lcm: 2 (if $n \ge 3$), number: $(n + 3)/2$ for $n$ odd, $(n + 6)/2$ for $n$ even, sum of squares: $2n$
Schur index values of irreducible representations over a splitting field 1 (all of them)
condition for a field to be a splitting field First, the field should have characteristic not equal to 2 or any prime divisor of $n$. Also, take the cyclotomic polynomial $\Phi_n(x)$. Let $\zeta$ be a root of the polynomial. Then, the field should contain the element $\zeta + \zeta^{-1}$, i.e., the minimal polynomial for $\zeta + \zeta^{-1}$ should split completely.
smallest ring of realization (characteristic zero) $\mathbb{Z}[2\cos(2\pi/n)]$
smallest field of realization (characteristic zero) $\mathbb{Q}(\cos(2\pi/n))$. Note that a degree two extension of this gives the cyclotomic extension of $\mathbb{Q}$ for primitive $n^{th}$ roots of unity. The given field can be thought of as the intersection of the cyclotomic extension and the real numbers.
smallest size splitting field unclear. Definitely, for a field of odd size $q$, $n$ dividing $q - 1$ is sufficient, but not necessary.
degrees of irreducible representations over rational numbers PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Supergroups

Groups having the dihedral group as quotient

The dicyclic group, also called the binary dihedral group, of order $4n$, has the dihedral group of order $2n$ as a quotient -- in fact the quotient by its center, which is a cyclic subgroup of order two. the presentation for the dicyclic group is given by:

$\langle a,x \mid a^n = x^2 = (ax)^2 \rangle$.

Dicyclic groups whose order is a power of $2$ are termed generalized quaternion groups.