Dihedral Group of Order 6 - Permutations of A Set of Three Objects

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, 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 (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 is built up from the basic actions a and b, we say that the set {a,b} generates it.

The group has presentation

, also written

or

, also written

where a and b are swaps and r is a cyclic permutation.