Symmetric group of degree 3
In mathematics, the smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 (or D6; both are used) and the symmetric group of degree 3, with notation S3.
This page illustrates many group concepts using this group as example.
In two dimensions, the group D3 is the symmetry group of an equilateral triangle. In contrast with the case of a square or other polygon, all permutations of the vertices can be achieved by rotation and flipping over (or reflecting).
In three dimensions, there are two different symmetry groups which are algebraically the group D3:
Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block".
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:
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. Similarly,
so each of the above actions has an inverse.
By inspection, we can also determine associativity and closure; note for example that
The group is non-abelian since, for example, ab ≠ ba. Since it is built up from the basic actions a and b, we say that the set {a, b} generates it.
The group has presentation
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