In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...

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...

In abstract algebra, the center of a group G is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning...

representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which...

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties...

The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical...

Muted Group Theory (MGT) is a communication theory developed by cultural anthropologist Edwin Ardener and feminist scholar Shirley Ardener in 1975, that...

In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data...

finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called...

In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of...

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector...

In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It...

arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete...

mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra:...

abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally...

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...

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core...

discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential...

to symmetry groups such as the Euclidean group and the group of projective transformations. In 1874 Lie introduced the theory of Lie groups, aiming for...

mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the...

in quantum field theory. Stueckelberg and Petermann opened the field conceptually. They noted that renormalization exhibits a group of transformations...

foundations for combinatorial group theory. The following table lists some examples of presentations for commonly studied groups. Note that in each case there...

representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be...

split-complex numbers taking the transpose in a matrix ring. In group theory, an element of a group is an involution if it has order 2; that is, an involution...

group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups. Furthermore, representation theory is...

In group theory, the normal closure of a subset S {\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle...

representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly...

such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group G {\displaystyle...

are different definitions used in group theory and ring theory. The commutator of two elements, g and h, of a group G, is the element [g, h] = g−1h−1gh...

group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group,...