The isomorphism theorems were formulated in some generality for homomorphisms of modules by
Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in
Mathematische Annalen. Less general versions of these theorems can be found in work of
Richard Dedekind and previous papers by Noether.
Three years later,
B.L. van der Waerden published his influential Moderne Algebra, the first
abstract algebra textbook that took the
groups-
rings-
fields approach to the subject. Van der Waerden credited lectures by Noether on
group theory and
Emil Artin on algebra, as well as a seminar conducted by Artin,
Wilhelm Blaschke,
Otto Schreier, and van der Waerden himself on
ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.
Groups
We first present the isomorphism theorems of the
groups.
Technically, it is not necessary for to be a normal subgroup, as long as is a subgroup of the
normalizer of in . In this case, is not a normal subgroup of , but is still a normal subgroup of the product .
This theorem is sometimes called the second isomorphism theorem,[1]diamond theorem[2] or the parallelogram theorem.[3]
If is a subgroup of such that , then has a subgroup isomorphic to .
Every subgroup of is of the form for some subgroup of such that .
If is a normal subgroup of such that , then has a normal subgroup isomorphic to .
Every normal subgroup of is of the form for some normal subgroup of such that .
If is a normal subgroup of such that , then the quotient group is isomorphic to .
The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem.
Let be a group, and a normal subgroup of .
The canonical projection homomorphism defines a bijective correspondence
between the set of subgroups of containing and the set of (all) subgroups of . Under this correspondence normal subgroups correspond to normal subgroups.
This theorem is sometimes called the
correspondence theorem, the lattice theorem, and the fourth isomorphism theorem.
The
Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[4]
Discussion
The first isomorphism theorem can be expressed in
category theoretical language by saying that the
category of groups is (normal epi, mono)-factorizable; in other words, the
normal epimorphisms and the
monomorphisms form a
factorization system for the
category. This is captured in the
commutative diagram in the margin, which shows the
objects and
morphisms whose existence can be deduced from the morphism . The diagram shows that every morphism in the category of groups has a
kernel in the category theoretical sense; the arbitrary morphism f factors into , where ι is a monomorphism and π is an epimorphism (in a
conormal category, all epimorphisms are normal). This is represented in the diagram by an object and a monomorphism (kernels are always monomorphisms), which complete the
short exact sequence running from the lower left to the upper right of the diagram. The use of the
exact sequence convention saves us from having to draw the
zero morphisms from to and .
If the sequence is right split (i.e., there is a morphism σ that maps to a π-preimage of itself), then G is the
semidirect product of the normal subgroup and the subgroup . If it is left split (i.e., there exists some such that ), then it must also be right split, and is a
direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an
abelian category (such as
that of abelian groups), left splits and right splits are equivalent by the
splitting lemma, and a right split is sufficient to produce a
direct sum decomposition . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence .
In the second isomorphism theorem, the product SN is the
join of S and N in the
lattice of subgroups of G, while the intersection S ∩ N is the
meet.
The third isomorphism theorem is generalized by the
nine lemma to
abelian categories and more general maps between objects.
Note on numbers and names
Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.
Comparison of the names of the group isomorphism theorems
It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, as one of isomorphism theorems, but when included, it is the last one.
Rings
The statements of the theorems for
rings are similar, with the notion of a normal subgroup replaced by the notion of an
ideal.
In particular, if is surjective then is isomorphic to .[15]
Theorem B (rings)
Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:
The
sumS + I = {s + i | s ∈ S, i ∈ I } is a subring of R,
The intersection S ∩ I is an ideal of S, and
The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.
Theorem C (rings)
Let R be a ring, and I an ideal of R. Then
If is a subring of such that , then is a subring of .
Every subring of is of the form for some subring of such that .
If is an ideal of such that , then is an ideal of .
Every ideal of is of the form for some ideal of such that .
If is an ideal of such that , then the quotient ring is isomorphic to .
Theorem D (rings)
Let be an ideal of . The correspondence is an
inclusion-preserving
bijection between the set of subrings of that contain and the set of subrings of . Furthermore, (a subring containing ) is an ideal of if and only if is an ideal of .[16]
In particular, if φ is surjective then N is isomorphic to M / ker(φ).
Theorem B (modules)
Let M be a module, and let S and T be submodules of M. Then:
The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M,
The intersection S ∩ T is a submodule of M, and
The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.
Theorem C (modules)
Let M be a module, T a submodule of M.
If is a submodule of such that , then is a submodule of .
Every submodule of is of the form for some submodule of such that .
If is a submodule of such that , then the quotient module is isomorphic to .
Theorem D (modules)
Let be a module, a submodule of . There is a bijection between the submodules of that contain and the submodules of . The correspondence is given by for all . This correspondence commutes with the processes of taking sums and intersections (i.e., is a
lattice isomorphism between the lattice of submodules of and the lattice of submodules of that contain ).[17]
A congruence on an
algebra is an
equivalence relation that forms a subalgebra of considered as an algebra with componentwise operations. One can make the set of
equivalence classes into an algebra of the same type by defining the operations via representatives; this will be
well-defined since is a subalgebra of . The resulting structure is the
quotient algebra.
Theorem A (universal algebra)
Let be an algebra
homomorphism. Then the image of is a subalgebra of , the relation given by (i.e. the
kernel of ) is a congruence on , and the algebras and are
isomorphic. (Note that in the case of a group, iff, so one recovers the notion of kernel used in group theory in this case.)
Theorem B (universal algebra)
Given an algebra , a subalgebra of , and a congruence on , let be the trace of in and the collection of equivalence classes that intersect . Then
is a congruence on ,
is a subalgebra of , and
the algebra is isomorphic to the algebra .
Theorem C (universal algebra)
Let be an algebra and two congruence relations on such that . Then is a congruence on , and is isomorphic to
Theorem D (universal algebra)
Let be an algebra and denote the set of all congruences on . The set
is a
complete lattice ordered by inclusion.[18]
If is a congruence and we denote by the set of all congruences that contain (i.e. is a principal
filter in , moreover it is a sublattice), then
the map is a lattice isomorphism.[19][20]
Notes
^
abMilne (2013), Chap. 1, sec. Theorems concerning homomorphisms
Noether, Emmy, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern,
Mathematische Annalen96 (1927) pp. 26–61
McLarty, Colin, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". The Architecture of Modern Mathematics: Essays in history and philosophy (edited by
Jeremy Gray and José Ferreirós), Oxford University Press (2006) pp. 211–35.