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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. The branch of mathematics which study groups is called the group theory.

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:
You will often also see the axiom
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 nonabelian.
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:
However sometimes the group is thought of as analogous to addition and written additively:
Usually, only abelian groups are written additively.
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
Proof:
This group is also abelian: a + b = b + a.
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:
So we see that not every element of (Z,·) has an inverse and therefore, (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. (Remember to also verify that · is a binary operation on Q \ {0} by checking closure.)
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:
Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,
By inspection, we can also determine associativity and closure; note for example that
This group is called the symmetric group on 3 letters, or S_{3}. It has order 6 (or 3 factorial), and is nonabelian (since, for example, ab ≠ ba). Since S_{3} is built up from the basic actions a and b, we say that the set {a,b} generates it.
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.
Constructing new groups from given ones
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.
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