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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In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly, for each , the left-multiplication-by-g map sending each element x to gx is a permutation of G, and the map sending each element g to is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of .
Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Dihedral group of order 6 Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the operation "swap the first block and the second block", and b be the operation "swap the second block and the third block". We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position".
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element a, where an, the n-th power of an element a is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, e; the lowest such power is the order of the cycle, the number of distinct elements in it.
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