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)
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:
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.
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:
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.
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:
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.
On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:
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.
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:
By inspection, we can also determine associativity and closure; note for example that
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.
For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.
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.