In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
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.
We first present the isomorphism theorems of the groups.
See also: Fundamental theorem on homomorphisms.
Let G and H be groups, and let f : G → H be a homomorphism. Then:
In particular, if f is surjective then H is isomorphic to G / ker(f).
This theorem is usually called the first isomorphism theorem.
Let
G
S
G
N
G
SN
G
N
SN
S\capN
S
(SN)/N
S/(S\capN)
N
S
N
G
N
G
N
SN
This theorem is sometimes called the second isomorphism theorem, diamond theorem[1] or the parallelogram theorem.[2]
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting
G=\operatorname{GL}2(C)
S=\operatorname{SL}2(C)
N
C x I=\left\{\left(\begin{smallmatrix}a&0\ 0&a\end{smallmatrix}\right):a\inC x \right\}
S\capN=\{\pmI\}
I
SN=\operatorname{GL}2(C)
\operatorname{PGL}2(C):=\operatorname{GL}2\left(C)/(C x I\right)\cong\operatorname{SL}2(C)/\{\pmI\}=:\operatorname{PSL}2(C)
Let
G
N
G
K
G
N\subseteqK\subseteqG
G/N
K/N
G/N
K/N
K
G
N\subseteqK\subseteqG
K
G
N\subseteqK\subseteqG
G/N
K/N
G/N
K/N
K
G
N\subseteqK\subseteqG
K
G
N\subseteqK\subseteqG
(G/N)/(K/N)
G/K
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.
See main article: Lattice theorem. Let
G
N
G
G → G/N
G
N
G/N
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.[3]
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
f:G → H
\iota\circ\pi
\kerf
\kappa:\kerf → G
\kerf
H
G/\kerf
If the sequence is right split (i.e., there is a morphism σ that maps
G/\operatorname{ker}f
\operatorname{im}\kappa
\operatorname{im}\sigma
\rho:G → \operatorname{ker}f
\rho\circ\kappa=\operatorname{id}ker
\operatorname{im}\kappa x \operatorname{im}\sigma
\operatorname{im}\kappa ⊕ \operatorname{im}\sigma
0 → G/\operatorname{ker}f → H → \operatorname{coker}f → 0
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.
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.
| Comment | Author | Theorem A | Theorem B | Theorem C |
|---|---|---|---|---|
| No "third" theorem | Jacobson[4] | Fundamental theorem of homomorphisms | (Second isomorphism theorem) | "often called the first isomorphism theorem" |
| van der Waerden,[5] Durbin | Fundamental theorem of homomorphisms | First isomorphism theorem | Second isomorphism theorem | |
| Knapp[6] | (No name) | Second isomorphism theorem | First isomorphism theorem | |
| Grillet[7] | Homomorphism theorem | Second isomorphism theorem | First isomorphism theorem | |
| Three numbered theorems | (Other convention per Grillet) | First isomorphism theorem | Third isomorphism theorem | Second isomorphism theorem |
| Rotman[8] | First isomorphism theorem | Second isomorphism theorem | Third isomorphism theorem | |
| Fraleigh[9] | Fundamental homomorphism theorem or first isomorphism theorem | Second isomorphism theorem | Third isomorphism theorem | |
| Dummit & Foote[10] | First isomorphism theorem | Second or Diamond isomorphism theorem | Third isomorphism theorem | |
| No numbering | Milne[11] | Homomorphism theorem | Isomorphism theorem | Correspondence theorem |
| Scott[12] | Homomorphism theorem | Isomorphism theorem | Freshman theorem |
The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.
Let
R
S
\varphi:R → S
\varphi
R
\varphi
S
\varphi
R/\ker\varphi
\varphi
S
R/\ker\varphi
Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:
Let R be a ring, and I an ideal of R. Then
A
R
I\subseteqA\subseteqR
A/I
R/I
R/I
A/I
A
R
I\subseteqA\subseteqR
J
R
I\subseteqJ\subseteqR
J/I
R/I
R/I
J/I
J
R
I\subseteqJ\subseteqR
J
R
I\subseteqJ\subseteqR
(R/I)/(J/I)
R/J
Let
I
R
A\leftrightarrowA/I
A
R
I
R/I
A
I
R
A/I
R/I
The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over
Z
In the following, "module" will mean "R-module" for some fixed ring R.
Let M and N be modules, and let φ : M → N be a module homomorphism. Then:
In particular, if φ is surjective then N is isomorphic to M / ker(φ).
Let M be a module, and let S and T be submodules of M. Then:
Let M be a module, T a submodule of M.
S
M
T\subseteqS\subseteqM
S/T
M/T
M/T
S/T
S
M
T\subseteqS\subseteqM
S
M
T\subseteqS\subseteqM
(M/T)/(S/T)
M/S
Let
M
N
M
M
N
M/N
A\leftrightarrowA/N
A\supseteqN
M/N
M
N
To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.
A
\Phi\subseteqA x A
A x A
A/\Phi
\Phi
A x A
Let
f:A → B
f
B
\Phi:f(x)=f(y)
f
A
A/\Phi
\operatorname{im}f
f(x)=f(y)
f(xy-1)=1
Given an algebra
A
B
A
\Phi
A
\PhiB=\Phi\cap(B x B)
\Phi
B
[B]\Phi=\{K\inA/\Phi:K\capB ≠ \emptyset\}
B
\PhiB
B
[B]\Phi
A/\Phi
[B]\Phi
B/\PhiB
Let
A
\Phi,\Psi
A
\Psi\subseteq\Phi
\Phi/\Psi=\{([a']\Psi,[a'']\Psi):(a',a'')\in\Phi\}=[ ]\Psi\circ\Phi\circ
| -1 | |
| [ ] | |
| \Psi |
A/\Psi
A/\Phi
(A/\Psi)/(\Phi/\Psi).
Let
A
\operatorname{Con}A
A
\operatorname{Con}A
\Phi\in\operatorname{Con}A
\left[\Phi,A x A\right]\subseteq\operatorname{Con}A
\Phi
\left[\Phi,A x A\right]
\operatorname{Con}A
\alpha:\left[\Phi,A x A\right]\to\operatorname{Con}(A/\Phi),\Psi\mapsto\Psi/\Phi