Examples of Groups - Permutations of A Set of Three Elements

Permutations of A Set of Three Elements

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". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:

• e : RGB → RGB
• a : RGB → GRB
• b : RGB → RBG
• ab : RGB → BRG
• ba : RGB → GBR
• aba : RGB → BGR

Note that aa has the effect RGB → GRB → RGB; so we can write aa = e. Similarly, bb = (aba)(aba) = e; (ab)(ba) = (ba)(ab) = e; so every element has an inverse.

By inspection, we can determine associativity and closure; note in particular that (ba)b = aba = b(ab).

Since it is built up from the basic operations a and b, we say that the set {a,b} generates this group. The group, called the symmetric group S₃, has order 6, and is non-abelian (since, for example, ab ≠ ba).