# 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:

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

• (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.

## 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)