Group (mathematics)

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory.

Related Topics:
Mathematics - Set - Binary operation - Group theory

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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. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms.

Related Topics:
Evariste Galois - 1830 - Algebraic equation - Radical - Permutation - Abelian group - Quadratic form

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A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. 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.

Related Topics:
Integer - Rational number - Real number - Complex number - Matrices

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Groups underlie many other algebraic structures such as fields and vector spaces. They are also important tools for studying symmetry in all its forms; the principle that the symmetries of any object form a group is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to mathematical physics (for example, in particle physics).

Related Topics:
Algebraic structure - Field - Vector space - Symmetry - Mathematical physics - Particle physics

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