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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.
A p (G) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group [,]): G/A p (G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.
The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.
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.
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of
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.