Dihedral group

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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 D_n, sometimes as \operatorname{Dih}_n, and sometimes as D_{2n} (this wiki uses D_{2n}) is defined in the following equivalent ways:

\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.

Related notions

Generalizations

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 False for n \ge 3 For n \ge 3, the elements a,x do not commute.
Nilpotent group True for n a power of 2, false otherwise
Metacyclic group True
Supersolvable group True
Solvable group True
T-group True for n odd or twice an odd number, false for n a multiple of 4
Rational group True for n \le 4, false for n \ge 5
Rational-representation group True for n \le 4, false for n \ge 5
Ambivalent group True Elements in the cyclic subgroup are conjugate via x, elements outside have order two.

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

Further information: Subgroup structure of dihedral groups

There are two kinds of subgroups:

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

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.


Internal links

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More 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)