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A class of groups is a set-theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativity ).
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. Groups recur throughout mathematics ...
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...
class function A class function on a group G is a function that it is constant on the conjugacy classes of G. class number The class number of a group is the number of its conjugacy classes. commutator The commutator of two elements g and h of a group G is the element [g, h] = g −1 h −1 gh. Some authors define the commutator as [g, h] = ghg ...
In group theory, a branch of mathematics, a formation is a class of groups closed under taking images and such that if G/M and G/N are in the formation then so is G/M∩N. Gaschütz (1962) introduced formations to unify the theory of Hall subgroups and Carter subgroups of finite solvable groups.
If G is a one-relator group and is a Magnus subgroup then the subgroup membership problem for H in G is decidable. [10] It is unknown if one-relator groups have solvable conjugacy problem. It is unknown if the isomorphism problem is decidable for the class of one-relator groups.
Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Carl Friedrich Gauss, a composition law was defined on certain equivalence classes of forms.