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2 The group of translations of the plane 3 A symmetry group 4 Matrix groups 5 Free group on two generators 6 Some more finite groups |
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:
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 o b as follows: first follow the prescription of b, then that of a.
For instance, if
The set of all translations of the plane with composition as operation forms a group:
Permutations of a set
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.The group of translations of the plane
and
then
(see Pythagorean theorem for why this is so, geometrically).
This is an Abelian group and our first (nondiscrete) example of a Lie group: a group whose underlying set is 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 square concrete slab of a certain thickness.
In order to describe its symmetry, we form the set of all those rigid movements of the slab that don't make a visible difference.
For instance, if you turn it by 90 degrees clockwise, then it still looks the same, so this movement is one element of our set, let's call it R.
We could also flip the slab horizontally so that its underside become up.
Again, after performing this movement, the slab looks the same, so this is also an element of our set and we call it T.
Then there's of course the movement that does nothing; it's denoted by I.
Now if you have two such movements a and b, you can define the composition a o b as above: you first perform the movement b and then the movement a.
The result will leave the slab looking like before.
The point is that the set of all those movements, with composition as operation, forms a group.
This group is the most concise description of the slab's symmetry.
Chemists use symmetry groups of this type to describe the symmetry of crystals.
Let's investigate our slab symmetry group some more.
Right now, we have the elements R, T and I, but we can easily form more:
for instance R o R, also written as R2, is a 180 degree turn (clockwise or counter-clockwise doesn't matter).
R3 is a 270 degree clockwise rotation, or, what is the same thing, a 90 degree counter-clockwise rotation.
We also see that T2 = I and also R4 = I.
Here's an interesting one: what does R o T do?
First flip horizontally, then rotate.
Try to visualize that R o T = T o R3.
Also, R2 o T is a vertical flip and is equal to T o R2.
This group is actually finite (it has order 8), and we can record everything there is to know about it in a multiplication table:
A symmetry group
| o | I | T | R | R2 | R3 | RT | R2T | R3T |
|---|---|---|---|---|---|---|---|---|
| I | I | T | R | R2 | R3 | RT | R2T | R3T |
| T | T | I | R3T | R2T | RT | R3 | R2 | R |
| R | R | RT | R2 | R3 | I | R2T | R3T | T |
| R2 | R2 | R2T | R3 | I | R | R3T | T | RT |
| R3 | R3 | R3T | I | R | R2 | T | RT | R2T |
| RT | RT | R | T | R3T | R2T | I | R3 | R2 |
| R2T | R2T | R2 | RT | T | R3T | R | I | R3 |
| R3T | R3T | R3 | R2T | RT | T | R2 | R | I |
For any two elements in the group, the table records what their composition is. Note how every element appears in every row and every column exactly once; this is not a coincidence. You may want to verify some entries. Here we wrote "R3T" as a short hand for R3 o T.
Mathematicians know this group as the dihedral group of order 8, and call it either D4 or D8, depending on what textbook they learned from in graduate school.
This was our first example of a non-abelian group: the operation o here is not commutative, which you can see from the table; the table is not symmetrical about the main diagonal.
If n is some positive integer, we can consider the set of all invertible n by n matrices over the reals, say.
This is a group with matrix multiplication as operation. It is called the general linear group, GL(n).
Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin).
If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group, SL(n).
Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space.
If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group O(n).
Geometrically, this consists of all combinations of rotations and reflections that fix the origin.
These are precisely the transformations which preserve lengths and angles.
Finally, if we impose both restrictions, then we get the special orthogonal group SO(n), which consists of rotations only.
These groups are our first examples of infinite non-abelian groups.
They are also Lie groups.
If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n).
We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n).
Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups.
The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a-1, b and b-1 such that no a appears directly next to an a-1 and no b appears directly next to an b-1.
Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string.
For instance: "abab-1a-1" concatenated with
"abab-1a" yields "abab-1a-1abab-1a", which gets reduced to "abaab-1a".
One can check that the set of those strings with this operation forms a group with neutral element the empty string ε := "".
(Usually the quotation marks are left off, which is why you need the symbol ε!)
This is another infinite non-abelian group.
Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach-Tarski paradox.
See the list of small groups for some more examples.Matrix groups
Free group on two generators
Some more finite groups