Dihedral group of order 6 Totally Explained

Everything about Dihedral Group Of Order 6 totally explained

The smallest non-abelian group has 6 elements. It is a dihedral group with notation D₃ or D₆ (unfortunately both are used) and the symmetric group of degree 3, with notation S₃. This page illustrates many group concepts using this group as example.

Symmetry groups

In 2D the group D₃ is the symmetry group of an equilateral triangle. As opposed to the case of for example a square, all permutations of the vertices can be achieved by rotation and flipping over (or reflecting).
In 3D there are two different symmetry groups which are algebraically the group D₃:

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): C_3v

Permutations of a set of three objects

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, for example, "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 (RBG)

ba : RGB → GBR or (RGB)

aba : RGB → BGR or (RB) Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,

bb = e,

(aba)(aba) = e, and

(ab)(ba) = (ba)(ab) = e; so each of the above actions has an inverse. By inspection, we can also determine associativity and closure; note for example that

(ab)a = a(ba) = aba, and

(ba)b = b(ab) = aba. The group is non-abelian since, for example, ab ≠ ba. Since it's built up from the basic actions a and b, we say that the set . Thus the average is six, the number of orbits.

Further Information

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