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

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...

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

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is...

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

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

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...

abstract theories. For instance, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups...

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...

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...

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

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...

renormalization group flow the field theory is conformally invariant. As the scale varies, it is as if one is decreasing (as RG is a semi-group and doesn't...

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

theory — Galois theory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —...

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

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...

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

subspaces, is denoted SO+(p, q). The group O(3, 1) is the Lorentz group that is fundamental in relativity theory. Here the 3 corresponds to space coordinates...

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...

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

In the mathematical field of group theory, the transfer defines, given a group G and a subgroup H of finite index, a group homomorphism from G to the abelianization...

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

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

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...

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

simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group 2F4(2)′...

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