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Concept
Dihedral group of order 6
Summary
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.
This page illustrates many group concepts using this group as example.
The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders.
The dihedral group D3 is isomorphic to two other symmetry groups in three dimensions:
one with a 3-fold rotation axis and a perpendicular 2-fold rotation axis (hence three of these): D3
one with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C3v
Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The symmetric group S3 is then the group of all possible rearrangements of these blocks.
If we denote by a the action "swap the first two blocks", and by b the action "swap the last two blocks", we can write all possible permutations in terms of these two actions.
In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB ↦ RBG ↦ BRG, i.e., "take the last block and move it to the front".
If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
e : RGB ↦ RGB or ()
a : RGB ↦ GRB or (RG)
b : RGB ↦ RBG or (GB)
ab : RGB ↦ BRG or (RGB)
ba : RGB ↦ GBR or (RBG)
aba : RGB ↦ BGR or (RB)
The notation in brackets is the cycle notation.
Note that the action aa has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were; so we can write aa = e.
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