Group theory

thumb|Set inclusions between the [[natural numbers ]]

algebraic set with an invertible, associative internal operation admitting a neutral element

the basic algebraic structure of linear algebra; a module over a field, such that its elements can be added together or scaled by elements of the field

branch of mathematics that studies the algebraic properties of groups

simple and widely known encryption technique

system of algebraic operations defined for remainders under division by a fixed positive integer; system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus

In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.

a smooth projective curve of genus 1 equipped with a rational basepoint

function between groups that preserves multiplication structure

theorem that there exists a decomposition of a unit solid ball into a finite number of disjoint subsets, which can be put back together in a different way to yield two identical copies of the unit sphere

group homomorphism into the general linear group over a vector space

group obtained by aggregating similar elements of a larger group

operation of the elements of a group as transformations or automorphisms (mathematics)

thumb| is the group \mathbb{Z}/8\mathbb{Z}, the Integers modulo n|integers mod 8 under addition. The subgroup contains only 0 and 4. There are four left cosets of : itself, , , and (written using additive notation since this is the [[additive group). Together they partition the entire group into equal-size, non-overlapping sets. The index is 4.]]

group of transformations under which the object is invariant

theorem

problem of inverting exponentiation in finite groups

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group.

set of group elements of the form g⁻¹hg for fixed h and ranging over all g

the set of elements that commute with every element of a group

research program on the symmetries of geometry

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

in mathematics, an invertible element or a unit in a ring R

graph whose vertices and edges represent the elements of a group and their products with the generators of the group

Lie group of Lorentz transformations

bijective group homomorphism

Wikimedia article covering multiple topics

an element which, when added to any element x in the set, yields x

subset of a group such that all group elements can be expressed by finitely many group operations on its elements

group of geometric symmetries (isometries) that keep at least one point fixed

finite group with 8 elements, whose elements can be represented by multiplication of unit quaternions {±1, ±i, ±j, ±k}

smallest normal subgroup by which the quotient is commutative

group homomorphism from the symmetric group over a finite set to the two-element group {±1}

area in mathematics devoted to the study of finitely generated groups

automorphism of a group defined by conjugation of one of its elements

subgroup of a group G that each leaves invariant each element of a given subset of a G-set

Lie group of 3×3 upper triangular matrices under matrix multiplication

relations between power sums and elementary symmetric functions

measure of the relative magnitude to an overall mathematical group

Spanish mathematician

group for which a given group is a normal subgroup

intersection of all maximal proper subgroups

type of linear representation

category of groups and group homomorphisms

orientation-preserving mapping class group of the torus

connection between representation theory of the monster group and the modular j-invariant

in mathematics, a family of finite groups

aspect of history

group of which the group operation is to be thought of as addition

group such that every its subgroup is normal

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groups of point isometries in 3 dimensions

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: \mbox{SL}(2,\mathbf{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon a,b,c,d \in \mathbf{R}\mbox{ and }ad-bc=1\right\}.

convex and balanced set

function which returns the largest least common multiple of any partition of its input

algorithm for solving the discrete logarithm problem

two-dimensional group theory table

several notions

graph, constructed for a group

Group of all affine transformations of an affine space