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 with parameter $n$, denoted sometimes as $D_n$ and sometimes as $D_{2n}$ is defined in the following equivalent ways:

• It has the presentation:

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

• (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.

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.

Elements

In the case of twice an odd number

Suppose $n$ is odd and greater than $1$. Let $D_{2n}$ be the dihedral group of order $2n$. It has the following elements:

• $n$ elements of odd order, all of them in the cyclic subgroup $\langle a \rangle$ of order $n$. Of these, there are precisely $\varphi(d)$ elements of order $d$ for any divisor $d$ of $n$.
• $n$ elements of order $2$. These are the elements outside the cyclic subgroup $\langle a \rangle$.

The conjugacy classes of elements are as follows (a total of $(n+3)/2$ conjugacy classes):

• The identity element is its own conjugacy class.
• The non-identity elements in $\langle a \rangle$ occur in conjugacy classes of size two. Each element is conjugate in $D_{2n}$ to its inverse. Thus, these form $(n-1)/2$ conjugacy classes.
• The elements outside $\langle a \rangle$ all form a single conjugacy class of size $n$.

The equivalence classes of elements upto automorphism are as follows:

• Two elements inside $\langle a \rangle$ are related via an automorphism if and only if they generate the same cyclic subgroup.
• Any two elements outside $\langle a \rangle$ are related via an automorphism (in fact, they are in the same conjugacy class).

In the case of twice an even number

Suppose $n = 2m$, and $D_{2n}$ is the dihedral group of order $2n$. Then, $D_{2n}$ has the following conjugacy classes (a total of $(n + 6)/2$ conjugacy classes):

• There are $(n+2)/2$ conjugacy classes inside $\langle a \rangle$: The identity element and $a^m$ are both central elements. All other elements have a conjugacy class of size two: the element and its inverse.
• There are two conjugacy classes outside $a$, of size $n/2$ each.

Subgroups

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 $\tau(n)$.
• Subgroups of the form $\langle a^d, a^r x \rangle$, where $d|n$. The number of such subgroups equals the sum of all positive divisors of $n$, sometimes denoted $\sigma(n)$.

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	Size of regular polygon it acts on	Common name for the group	Comment
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

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 23-27, Section 1.2 Dihedral Groups (the entire section discusses dihedral groups from a number of perspectives)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 24 (definition introduced in paragraph)
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 78, Exercise 34 (a) (definition introduced in exercise)
• Topics in Algebra by I. N. Herstein, ^{More info}, Page 54, Problem 17
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 6 (definition introduced informally, in paragraph, using the geometric perspective)
• An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 42, under The symmetry group of the regular n-gon
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 50 (definition introduced as a subgroup of the symmetric group)