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

\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 978-0471433347More 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)