Dihedral group of order 8
Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.
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:
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).
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. If you have two such translations a and b, they 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
and
then
(see Pythagorean theorem for why this is so, geometrically).
The set of all translations of the plane with composition as operation forms a group:
This is an abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.
Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a glass square of a certain thickness (with a letter "F" written on it, just to make the different positions discriminable). In order to describe its symmetry, we form the set of all those rigid movements of the square that don't make a visible difference (except the "F"). For instance, if you turn it by 90° clockwise, then it still looks the same, so this movement is one element of our set, let's call it a. We could also flip it horizontally so that its underside become up. Again, after performing this movement, the glass square looks the same, so this is also an element of our set and we call it b. Then there's of course the movement that does nothing; it's denoted by e.
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