Math 2.0

Group theory and generalizations Definition

Dihedral group

Builds on Symmetry groupSubgroup generated by a subset

Definition

Let n≥3n \geq 3 be an integer. Let PP be a regular nn-gon in the plane with centre 00. The dihedral group DnD_n is the group of all isometries of the plane that map PP onto itself, with composition as the operation. Let rr be the rotation by the angle 2π/n2\pi/n about 00, and let ss be a reflection in a line of symmetry of the polygon.

Remarks

The elements of DnD_n are the rotations e,r,…,rn−1e, r, \dots, r^{n-1} and the reflections s,rs,…,rn−1ss, rs, \dots, r^{n-1}s. The relations rn=er^n = e, s2=es^2 = e and srs−1=r−1srs^{-1} = r^{-1} hold, and these relations give a presentation of DnD_n. In abstract algebra, many authors write D2nD_{2n} for this group, so the index is the order and not the number of sides. The group DnD_n is not abelian, and it is a semidirect product of ⟨r⟩\langle r \rangle and ⟨s⟩\langle s \rangle.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptSymmetry group, concept, no evidence levelSymmetry groupconceptSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionDihedral group, definition, no evidence levelDihedral groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Symmetry group concept
  3. Subgroup definition
  4. Subgroup criterion statement
  5. Intersection of subgroups is a subgroup statement
  6. Subgroup generated by a subset definition
  7. Dihedral group definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dihedral_group, Definition
  2. Web https://en.wikipedia.org/wiki/Dihedral_group, Other definitions
About this page
Tag09GK3DE0WV
URIm2://09GK3DE0WV
Kinddefinition
WritingW1 stated
MSC20B25
DefinesDihedral group
UsesSymmetry group, Subgroup generated by a subset

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by