Group (mathematics)

The mathematical concept of a group is one of the fundamental notions of modern algebra. Groups underlie the other algebraic structures such as fields and vector spaces and are also important tools for studying symmetry in all its forms.

Definition

A group is defined as a set, say G and a binary operation, say "*", denoted (G,*), such that

(Closure) for all a and b in G, a * b belong to G.
(Associativity) for all a, b and c in G, (a * b) * c = a * (b * c).
(Identity element) there is an element e in G such that for all a in G, e * a = a = a * e.
(Inverse element) for all a in G there is a b in G such that a * b = e = b * a.

From these axioms, one can immediately derive several important consequences, for example that a group has only one identity element and that every element has only one inverse. This falls under elementary group theory.

Usually the operation, whatever it really is, is denoted like multiplication; we write a * b for the product of a and b, 1 for the identity and a^-1 for the inverse of a.

A group is called Abelian if the operation * is commutative, i.e. if a * b = b * a for all a and b in G. In this case, the operation is often written as + instead of *, the identity element as 0, and the inverse of the element a is written as (-a).

A group is called finite if it has finitely many elements. In this case, the order of the group, denoted by |G| or o(G), is the number of elements of G.

Some basic examples

Integers

A group that we are introduced to in elementary school is the integers under addition. Thus, let Z be the set of integers={...,-4,-3,-2,-1,0,1,2,3,4,...} and let the symbol "+" indicate the operation of addition. Then, (Z,+) is a group.

Proof:

If a and b are integers then a + b is an integer: Closure.
If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
0 is an integer and for any integer a, a + 0 = a. (Z,+) has an identity element.
If a is an integer, then there is an integer b = (-a), such that a + b = 0. Every element of (Z,+) has an inverse.

Integers with multiplication

Question: Given the set of integers, Z, as above, and the operation multiplication, denoted by * is (Z,*) a group?

If a and b are integers then a*b is an integer. Closure.
If a, b, and c are integers, then (a * b) * c = a * (b * c). Associativity.
1 is an integer and for any integer a, a * 1 = a and 1 * a = a. (Z,*) has an identity element.
BUT, if a is an integer, there is not necessarily an integer b such that a * b = 1. There may be a rational number b like that, but not an integer.

So we see that not every element of (Z,*) has an inverse and therefore, (Z,*) is not a group. Instead it is only a monoid.

Rational numbers with multiplication

Question: Given the set of rational numbers Q, that is the set of numbers a / b such that a and b are integers and b is nonzero, and the operation multiplication, denoted by "*", is (Q,*) a group?

Answer: No. The rational number 0 is in Q, but does not have a multiplicative inverse. (Q,*) is a monoid but not a group.

If we modify the constraint so that neither a nor b may be 0 (i.e. the set is now Q excluding 0, denoted Q{0}) then (Q{0},*) does form a group. The inverse of a / b is b / a, and the other group axioms are simple to check.

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 aob 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).

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

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

Symmetry groups

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 group table:

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	R³	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.

This was our first example of a non-abelian group: the operation o here is not commutative, as the table is not symmetrical about the main diagonal.

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. Intuitively, it contains all combinations of rotations, reflections and dilations of n-dimensional space.

If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group. The elements, intuitively, are the rotations of n-dimensional space.

These two groups are our first examples of infinite non-abelian groups.

Free group in 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.

This is another infinite non-abelian group.

Free groups are important in algebraic topology; the free group in two generators is also used for the proof of the Banach-Tarski Paradox.

The theory of groups

Initially, one needs tools to compare groups (group homomorphisms) and to construct new groups from old ones (subgroups, normal subgroups, factor groups, center of a group, derived group, and product of groups, especially semidirect and direct product). When studying these concepts, one encounters the theorem of Lagrange, the fundamental theorem on homomorphisms, and the isomorphism theorems. An important tool here is the concept of a coset of a subgroup (see under subgroup).

For more detailed study of the lattice of subgroups of a given finite group, the notion of p-group and the Sylow theorems are useful. A helpful tool for proving these theorems is the concept of a group action.

The cyclic groups, groups that can be generated by a single element, can be completely characterized; they are all abelian. More generally, all finitely generated (and in particular the finite) abelian groups can be completely classified, a theorem which finds wide applications. (see finitely generated abelian group).

The situation is much more complicated when trying to get a handle on the finite non-abelian groups. Every finite group is built up from simple groups, and in a celebrated huge theorem, all finite simple groups have at last been classified (see classification of finite simple groups).

The solvable group and nilpotent groups are important because they appear prominently in Galois theory when trying to classify those polynomial equations that can be solved with radicals.

An important tool in group theory are group representations; one basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study.

In order to specify an abstract group, one often uses generators and relations (presentation of a group). One can then ask algorithmical questions about these groups, for instance: "are two groups isomorphic?", or "is this group trivial?". Several of these questions are unsolvable with a general algorithm, see for instance the word problem for groups.

Applications

Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of automorphism groups.

In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by a polynomial's roots.

In algebraic topology, groups are used to describe invariants of topological spaces. They are called invariants because they are defined in such a way that they don't change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

The concept of Lie group is important in the study of differential equations and manifolds; they "marry" analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis. Groupoids, which are objects similar to groups except that the composition a * b need not be defined for all a and b, also arise in the study of topological and analytical structures.

In chemistry, groups are used to classify crystal structures, regular polyhedra and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the natural laws are supposed to obey. Physicists are very interested in representations of groups, especially Lie groups, since these representations often point the way to the "possible" physical theories.

Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.

External links:

History of the abstract group concept