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# Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. [1]

59 relations: Abelian group, Abnormal subgroup, Abstract algebra, Ascendant subgroup, Automorphism, Évariste Galois, C-normal subgroup, Center (group theory), Central product, Centralizer and normalizer, Characteristic subgroup, Commutator subgroup, Complete lattice, Conjugacy class, Conjugate closure, Conjugate-permutable subgroup, Contranormal subgroup, Core (group theory), Coset, Dedekind group, Descendant subgroup, Dihedral group, Domain of a function, Euclidean group, Greatest and least elements, Group (mathematics), Group homomorphism, Ideal (ring theory), Imperfect group, Index of a subgroup, Inner automorphism, Isomorphism, Isomorphism theorems, Israel Nathan Herstein, John Wiley & Sons, Kernel (algebra), Lattice (order), Logical equivalence, Malnormal subgroup, Modular lattice, Modular subgroup, Orthogonal group, Paranormal subgroup, Perfect group, Polynormal subgroup, Pronormal subgroup, Quasinormal subgroup, Quotient group, Rubik's Cube group, Seminormal subgroup, ... Expand index (9 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Abnormal subgroup

In mathematics, in the field of group theory, an abnormal subgroup is a subgroup H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x, where Hx denotes the conjugate subgroup xHx-1.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Ascendant subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Évariste Galois

Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.

## C-normal subgroup

In mathematics, in the field of group theory, a subgroup H of a group G is called c-normal if there is a normal subgroup T of G such that HT.

## Center (group theory)

In abstract algebra, the center of a group,, is the set of elements that commute with every element of.

## Central product

In mathematics, especially in the field of group theory, the central product is way of producing a group from two smaller groups.

## Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition.

## Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group.

## Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

## Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

## Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

## Conjugate closure

In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the set of the conjugates of the elements of S: The conjugate closure of S is denoted G> or G. The conjugate closure of any subset S of a group G is always a normal subgroup of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains S. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection of all normal subgroups of G which contain S. Any normal subgroup is equal to its normal closure.

## Conjugate-permutable subgroup

In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups.

## Contranormal subgroup

In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group.

## Core (group theory)

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group.

## Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

## Dedekind group

In group theory, a Dedekind group is a group G such that every subgroup of G is normal.

## Descendant subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.

## Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Euclidean group

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.

## Greatest and least elements

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Imperfect group

In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients.

## Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

## Inner automorphism

In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.

## Israel Nathan Herstein

Israel Nathan Herstein (March 28, 1923 – February 9, 1988) was a mathematician, appointed as professor at the University of Chicago in 1951.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

## Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

## Malnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a group G is termed malnormal if for any x in G but not in H, H and xHx^ intersect in the identity element.

## Modular lattice

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:;Modular law: x ≤ b implies x ∨ (a ∧ b).

## Modular subgroup

In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Paranormal subgroup

In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup.

## Perfect group

In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial).

## Polynormal subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated.

## Pronormal subgroup

In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way.

## Quasinormal subgroup

In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups.

## Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

## Rubik's Cube group

The Rubik’s Cube group is a group (G, \cdot) that represents the structure of the Rubik's Cube mechanical puzzle.

## Seminormal subgroup

In mathematics, in the field of group theory, a subgroup A of a group G is termed seminormal if there is a subgroup B such that AB.

## Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Subnormal subgroup

In mathematics, in the field 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 the next, beginning at H and ending at G. In notation, H is k-subnormal in G if there are subgroups of G such that H_i is normal in H_ for each i. A subnormal subgroup is a subgroup that is k-subnormal for some positive integer k. Some facts about subnormal subgroups.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## T-group (mathematics)

In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal.

## Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

## Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

## References

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