Examples of groups

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". 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 = bab = 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). A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. Two translations such as a and b can be composed to form a new translation a ∘ b as follows: first follow the prescription of b, then that of a. For instance, if a = "move North-East for 3 miles" and b = "move South-East for 4 miles" then a ∘ b = "move to bearing 8.13° for 5 miles" (bearing is measured counterclockwise and from East) Or, if a = "move to bearing 36.87° for 3 miles" (bearing is measured counterclockwise and from East) and b = "move to bearing 306.