We first state the three isomorphism theorems in the context of groups.

Then and thus the result follows by use of the first isomorphism theorem.

The isomorphism theorems known from ordinary group theory are not always true in the topological setting.

The homomorphism theorem is used to prove the isomorphism theorems.

This follows readily from the isomorphism theorems for rings.

The isomorphism theorems for vector spaces and abelian groups are special cases of these.

These results are nothing more than a particularization of the first isomorphism theorem in universal algebra.

This result is sometimes called the diamond isomorphism theorem for modular lattices.

This approach is based on the isomorphism theorem for standard Borel spaces .

The question long defied analysis, but was finally and completely answered with the Ornstein isomorphism theorem.

She employed the concept in her original formulation of the three Noether isomorphism theorems.

In mathematics, the Ornstein isomorphism theorem is a deep result for ergodic theory.

In the following diagram expressing the first isomorphism theorem, commutativity means that :

It allows one to refine the first isomorphism theorem:

Putting these together, the octahedral axiom asserts the "third isomorphism theorem":

See quotient group and isomorphism theorem.

The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.

Countable discrete amenable groups obey the Ornstein isomorphism theorem.

Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule.

A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.

The norm residue isomorphism theorem implies the Quillen-Lichtenbaum conjecture.

The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples.

First isomorphism theorem, that describe the relationship between quotients, homomorphisms, and subobjects (mathematics)