Examples of groups

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Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.

[edit] Permutations of a set of three elements

Cycle graph for S₃ (or D₆). A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that ba²=ab and ba³=e, as well as the fact that ab²=ba and ab³=e The other "loops" are roots of unity so that, for example a²=e.

Main article: 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 = 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).

[edit] The group of translations of the plane

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

a = "move North-East for 2 miles"

and

b = "move South-East for 2 miles"

then

a ^o b = "move East for sqrt(8) miles"

(see Pythagorean theorem for why this is so, geometrically).

The set of all translations of the plane with composition as operation forms a group:

1. If a and b are translations, then a ^o b is also a translation.
2. Composition of translations is associative: (a ^o b) ^o c = a ^o (b ^o c).
3. The identity element for this group is the translation with prescription "move zero miles in whatever direction you like".
4. The inverse of a translation is given by walking in the opposite direction for the same distance.

This is an Abelian group and our first (nondiscrete) example of a Lie group: a group whose underlying set is a manifold.

[edit] The symmetry group of a square (dihedral group D₈ or D₄)

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, R^{2 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 Cayley table (multiplication table):

Cycle graph for D₄. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-a-a²-a³ loop reflects the fact that the successive powers of a are distinct until a⁴=e. This loop also reflects the fact that successive powers of a³ are distinct until (a³)⁴=e. The other "loops" are roots of the identity so that, for example b²=e. In the text of the article, R=a, T=b and I=e.

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 R^{3 o} T.

Mathematicians know this group as the dihedral group of order 8, and call it either D₄ or D₈ depending on what notation they use for dihedral groups. This was an 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.

The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13).

[edit] Matrix groups

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 happen to be Lie groups. In fact, most of the important Lie groups (but not all) can be expressed as matrix 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.

For example, we have the general linear groups over finite fields. The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121)

[edit] Free group on two generators

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.

[edit] The set of maps

[edit] The sets of maps from a set to a group

Let G be a group and S a nonempty set. The set of maps M(S,G) is itself a group; namely for two maps f,g of S into G we define fg to be the map such that (fg)(x)=f(x)g(x) for every x∈S and f ^-1 to be the map such that f ^-1(x)=f(x)^-1.

Take maps f, g, and h in M(S,G). For every x in S, f(x) and g(x) are both in G, and so is (fg)(x). Therefore fg is also in M(S,G), or M(S,G) is closed. For ((fg)h)(x)=(fg)(x)h(x)=(f(x)g(x))h(x)=f(x)(g(x)h(x))=f(x)(gh)(x)=(f(gh))(x), M(S,G) is associative. And there is a map i such that i(x)=e where e is the unit element of G. The map i makes all the functions f in M(S,G) such that if=fi=f, or i is the unit element of M(S,G). Thus, M(S,G) is actually a group.

If G is commutative, (fg)(x)=f(x)g(x)=g(x)f(x)=(gf)(x). Therefore so is M(S,G).

[edit] The groups of permutations

Let S be a nonempty set. Let G be the set of bijective mappings of S onto itself. Then G, also denoted by Perm(S), is a group with ordinary composition of mappings. The unit element of G is the identity map of S. For two maps f and g in G are bijective, fg is also bijective. Therefore G is closed. The composition of maps is associative, so is G. Thus, G is a group.

S may be either finite, or infinite.

[edit] Some more finite groups

See the list of small groups for some more examples.