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Concept
Examples of groups
Résumé
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.
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