Isomorphism theorems
Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures.
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.
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.
[3] An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting
, the group of invertible 2 × 2 complex matrices,
defines a bijective correspondence between the set of subgroups of
[4] 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
(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
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
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.
Here we give some examples of the group isomorphism theorems in the literature.
Notice that these theorems have analogs for rings and modules.
The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.
is an inclusion-preserving bijection between the set of subrings
This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of
[17] To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.
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
The resulting structure is the quotient algebra.
, so one recovers the notion of kernel used in group theory in this case.)
is a complete lattice ordered by inclusion.
