In mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory.
The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals.
A great many of the objects investigated in mathematics turn out to be groups, including familiar number systems, such as the integers, rational, real, and complex numbers under addition, non-zero rational, real, and complex numbers under multiplication, non-singular matricies under multiplication, invertable functions under composition, and so on. Group Theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right. Groups underlie the other algebraic structures such as fieldss and vector spaces and are also important tools for studying symmetry in all its forms. For these reasons, group theory is considered to be an important area in modern mathematics, and has many applications to mathematical physics (for example, in particle theory)
History
Basic definitions
A group (G,*) is defined as a set G together with a binary operation *: G × G → G. We write "a * b" for the result of applying the operation * to the two elements a and b of G. To have a group, * must satisfy the following axioms:
- 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 an element b in G such that a * b = e = b * a, where e is the identity element from the previous axiom.
- Closure: For all a and b in G, a * b belongs to G.
It should be noted that there is no requirement in a group that a * b = b * a (commutativity). A group in which this equation holds for all a and b in G, is called abelian (after the mathematican Niels Abel). Groups lacking this property are called non-abelian.
The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set.
Note that we often refer to the group (G,*) as simply "G", leaving the operation * unmentioned. But to be perfectly precise, different operations on the same set define different groups.
Notation for groups
Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:
- We write "a · b" or even "ab" for a * b and call it the product of a and b;
- We write "1" for the identity element and call it the unit element;
- We write "a^{−1}" for the inverse of a and call it the reciprocal of a.
- We write "a + b" for a * b and call it the sum of a and b;
- We write "0" for the identity element and call it the zero element;
- We write "−a" for the inverse of a and call it the opposite of a.
When being noncommital, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a^{−1} for the inverse of a.
If S is a subset of G, and x an element of G then in multiplicative notation, xS is the set of all products {xs} for s in S; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : for all s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets.
Some elementary examples and nonexamples
An abelian group: the integers under addition
A group that we are introduced to in elementary school is the integers under addition. For this example, 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 (written additively).
Proof:
- If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)
- If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
- 0 is an integer and for any integer a, 0 + a = a = a + 0. (Identity element)
- If a is an integer, then there is an integer b := −a, such that a + b = 0 = b + a. (Inverse element)
The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.
Not a group: the integers under multiplication
On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:
- If a and b are integers then a · b is an integer. (Closure; · really is a binary operation)
- If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)
- 1 is an integer and for any integer a, 1 · a = a = a · 1. (Identity element)
- But, if a is an integer, there is not necessarily an integer b such that a · b = 1 = b · a. There may be a rational number b like that, but not an integer. (Inverse element fails)
An abelian group: the nonzero rational numbers under multiplication
Consider 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 "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group.
However, if we instead use the set Q \\ {0} instead of Q, that is include every rational number except zero, then (Q \\ {0},·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero.
Just as the integers form a ring, so the rational numbers form the algebraic structure of a field. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.
A finite nonabelian group: permutations of a set
In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
- e : RGB → RGB
- a : RGB → GRB
- b : RGB → RBG
- ab : RGB → BRG
- ba : RGB → GBR
- aba : RGB → BGR
- bb = e,
- (aba)(aba) = e, and
- (ab)(ba) = (ba)(ab) = e;
By inspection, we can also determine associativity and closure; note for example that
- (ab)a = a(ba) = aba, and
- (ba)b = b(ab) = aba.
Every group can be expressed in terms of permutation groups like S_{3}; this result is Cayley's theorem and is studied as part of the subject of group actions.
Further examples
For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.
Simple theorems
- A group has exactly one identity element.
- Every element has exactly one inverse.
- You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b.
- The expression "a_{1} * a_{2} * ··· * a_{n}" is unambiguous, because the result will be the same no matter where we place parentheses.
- The inverse of a product is the product of the inverses in the opposite order: (a * b)^{−1} = b^{−1} * a^{−1}.
Constructing new groups from given ones
- If a subset H of a group (G,*) together with the operation * restricted on H is itself a group, then it is called a subgroup of (G,*).
- The direct sum of two groups (G,*) and (H,•) is the set G×H together with the operation (g_{1},h_{1})(g_{2},h_{2}) = (g_{1}*g_{2},h_{1}•h_{2}).
- Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.
Related topics
See Glossary of group theory for more definitions in group theory.
See elementary group theory for a list of elementary theorems in group theory.
See List of group theory topics for a list of all group theory topics covered in Wikipedia.