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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. [25] For example, repeated applications of the associativity axiom show that the unambiguity of a ⋅ b ⋅ c = ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b ...
The group consisting of all permutations of a set M is the symmetric group of M. p-group If p is a prime number, then a p-group is one in which the order of every element is a power of p. A finite group is a p-group if and only if the order of the group is a power of p. p-subgroup A subgroup that is also a p-group.
In mathematics, the category Grp (or Gp [1]) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory.
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with x p = e, where e is the identity element of G.
The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if G is any finitely generated group of exponent n, then G is a homomorphic image of B(m, n), where m is the number of generators of G. The Burnside problem for groups with bounded exponent can now be ...