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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.
Grigorchuk group. In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth.
Word problem for groups. In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group is the algorithmic problem of deciding whether two words in the generators represent the same element of . The word problem is a well-known example of an undecidable problem.
e. 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 xp = e, where e is the identity element of G.
The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). [1] The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by S n, and may be called the symmetric group on n letters. By Cayley's theorem, every group is isomorphic to some ...
Rank of a group. In the mathematical subject of group theory, the rank of a group G, denoted rank (G), can refer to the smallest cardinality of a generating set for G, that is. If G is a finitely generated group, then the rank of G is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of ...
In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, [1] or even in every group. [2] Words play an important role in the theory of ...
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.