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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. Before that groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms.

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, invertible 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 fields 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).

1 History

See Group theory.

2 Basic definitions

A group (G,*) is a nonempty 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:

AssociativityAbstract algebra Algebra In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are requir: For all a, b and c in G, (a * b) * c = a * (b * c).
Identity elementIn mathematics, an identity element (or neutral element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. The term identity element is often shortened to ident: There is an element e in G such that for all a in G, e * a = a = a * e.
Inverse elementIn mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combinati: 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.

You will often also see the axiom

Closure: For all a and b in G, a * b belongs to G.

The way that the definition above is phrased, this axiom isn't necessary, since binary operations are already required to satisfy closure. When determining if * is a group operation, however, it is nonetheless necessary to verify that * satisfies closure; this is part of verifying that it is in fact a binary operation.

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.

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