group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the correspondence theorem (also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem ) states that if

N

is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

G

, then there exists a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

from the set of all

subgroups 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 ∗. More precisely, ''H'' is a subgro ...

A

G

containing

N

, onto the set of all subgroups of the

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 some of the group structure (the rest of the structure is "factored" out). For examp ...

G/N

. The structure of the subgroups of

G/N

is exactly the same as the structure of the subgroups of

G

containing

N

, with

N

collapsed to the

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

. Specifically, if : ''G'' is a group, :

N \triangleleft G

, a

of ''G'', :

\mathcal = \

, the set of all subgroups ''A'' of ''G'' that contain ''N'', and :

\mathcal = \

, the set of all subgroups of ''G''/''N'', then there is a bijective map

\phi: \mathcal \to \mathcal

such that :

\phi(A) = A/N

for all

A \in \mathcal.

One further has that if ''A'' and ''B'' are in

\mathcal

then *

A \subseteq B

if and only if

A/N \subseteq B/N

; * if

A \subseteq B

then

, B:A,  = , B/N:A/N,

, where

, B:A,

is the

index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...

of ''A'' in ''B'' (the number of

coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

s ''bA'' of ''A'' in ''B''); *

\langle A,B \rangle / N = \left\langle A/N, B/N \right\rangle,

where

\langle A, B \rangle

is the subgroup of

G

generated by

A\cup B;

(A \cap B)/N = A/N \cap B/N

, and *

A

is a normal subgroup of

G

if and only if

A/N

is a normal subgroup of

G/N

. This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group. More generally, there is a monotone Galois connection

(f^*, f_*)

between the

lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, a ...

G

(not necessarily containing

N

) and the lattice of subgroups of

G/N

: the lower adjoint of a subgroup

H

G

is given by

f^*(H) = HN/N

and the upper adjoint of a subgroup

K/N

G/N

is a given by

f_*(K/N) = K

. The associated

closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...

on subgroups of

G

\bar H = HN

; the associated

kernel operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...

on subgroups of

G/N

is the identity. A proof of the correspondence theorem can be foun
here
Similar results hold for

rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

s, and

algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...

References

{{reflist Isomorphism theorems

See also

References