Wikipedia / Isomorphism theorems
Wikipedia / Isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruencess.
History
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. Three years later, B. L. van der Waerden published Moderne Algebra, an influential early abstract algebra textbook that helped standardize the structural treatment of groups, rings, and fields in which these theorems appear prominently.
Groups
We first present the isomorphism theorems of the groups.
Theorem A (groups)
be a homomorphism. Then:
The kernel is a normal subgroup,
is isomorphic to the quotient group.
is surjective.
This theorem is usually called the first isomorphism theorem.
Theorem B (groups)
. Then the following hold:
The product,
,
The intersection, and
are isomorphic.
is a subgroup of the normalizer.
This theorem is sometimes called the second isomorphism theorem, diamond theorem or the parallelogram theorem.
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line, the group of invertible 2 × 2 complex matrices, the subgroup of determinant is the identity matrix. Then the second isomorphism theorem states that:
Theorem C (groups)
. Then
.
.
.
.
.
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.
Theorem D (groups)
. 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.
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. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f, where ι is a monomorphism and π is an epimorphism (in a conormal category (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.
If the sequence is right split (i.e., there is a morphism σ to a -preimage of itself), then G is the semidirect product 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.
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.
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 a two-sided ideal. All statements hold if the rings involved are assumed to have a multiplicative identity, with homomorphisms preserving the identity, or if rings are not assumed to have an identity.
Theorem A (rings)
be a ring homomorphism. Then:
The kernel,
is isomorphic to the quotient ring.
.
Theorem B (rings)
. Then:
The sum,
, and
are isomorphic.
Theorem C (rings)
. Then
.
.
.
.
.
Theorem D (rings)
is an inclusion-preserving bijection and the set of subrings of .
Modules
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) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem.
In the following, "module" will mean either "left R-module" or "right R-module" for some fixed ring R.
Theorem A (modules)
be a module homomorphism. Then:
The kernel,
The image, and
is isomorphic to the quotient module.
.
Theorem B (modules)
. Then:
,
, and
are isomorphic.
Theorem C (modules)
Let M be a module, T a submodule of M.
.
.
.
Theorem D (modules)
. This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism).
Universal algebra
To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.
A congruence on an algebra is an equivalence relation 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. The resulting structure is the quotient algebra.
Theorem A (universal algebra)
be an algebra homomorphism (i.e. the kernel are isomorphic iff, so one recovers the notion of kernel used in group theory in this case.)
Theorem B (universal algebra)
. Then
,
, and
.
Theorem C (universal algebra)
Theorem D (universal algebra)
is a complete lattice ordered by inclusion. is a principal filter is a lattice isomorphism.
Notes
References
Cohn, Paul M., Universal algebra, Chapter II.3 p.57