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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.
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.
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 ...
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.
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 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.
A group may have a small base compared to the set it acts on. In the "best case", a base can have size 1, as in the case of the additive group of the integers . On the other hand, the symmetric groups and alternating groups have large bases (the symmetric group S n has base size n − 1), and there are often specialized algorithms that deal ...
If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that a m = e, where e denotes the identity element of the group, and a m denotes the product of m copies of a. If no such m exists, the order of a is infinite.