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...

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

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

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

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...

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

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...

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...

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

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...

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...

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

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...

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

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

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...

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...

between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory). Let R {\displaystyle...

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...

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...

groups, rings, and fields, based on the number of operations they use and the laws they follow, called axioms. Universal algebra and category theory provide...

representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field...

examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose...

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

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