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...

The term normal closure is used in two senses in mathematics: In group theory, the normal closure of a subset of a group is the smallest normal subgroup...

from that language. In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set...

smallest normal subgroup that contains each element of R. (This subgroup is called the normal closure N of R in F S {\displaystyle F_{S}} .) The group ⟨ S...

abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

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...

functions Normal function, in set theory Normal invariants, in geometric topology Normal matrix, a matrix that commutes with its conjugate transpose Normal measure...

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

contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:...

topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can...

with the field of fractions K, L a finite normal extension of K, B the integral closure of A in L. Then the group G = Gal ⁡ ( L / K ) {\displaystyle G=\operatorname...

members of the group of which it is a part. In other words, a subgroup N {\displaystyle N} of the group G {\displaystyle G} is normal in G {\displaystyle...

"locale"). Filters and nets are notions closely related to order theory and the closure operator of sets can be used to define a topology. Beyond these...

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,...

Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields...

of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in...

classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder...

that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include...

index in its normal closure. Scott (1987), 15.1.1, p. 441. Scott (1987), 15.1.2, p. 441. Scott, W. R. (1987), "15.1 FC groups", Group Theory, Dover, pp...

In group theory, the correspondence theorem (also the lattice theorem, and variously and ambiguously the third and fourth isomorphism theorem) states 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...

roots. Galois extension A normal, separable field extension. Primary extension An extension E/F such that the algebraic closure of F in E is purely inseparable...

subgroup of G if its normal closure is G itself. cyclic group A cyclic group is a group that is generated by a single element, that is, a group such that there...

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

{\displaystyle H^{K}} . Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups. Here are some facts relating...

factorisation Syntactic monoid Structure Group (mathematics) Lagrange's theorem (group theory) Subgroup Coset Normal subgroup Characteristic subgroup Centralizer...

In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle...

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

transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operator is usually called transitive closure logic...

change the orientations of blocks. This group is a normal subgroup of G. It can be represented as the normal closure of some moves that flip a few edges or...