Third isomorphism theorem

Third isomorphism theorem

The third isomorphism theorem is a fundamental result in group theory describing the relationship between successive quotients by nested normal subgroups. It appears in texts alongside the first isomorphism theorem, second isomorphism theorem, and the correspondence theorem and underpins structural results used in the study of Galois groups, Lie algebras, and module theory in the tradition of Hilbert-style algebra. The theorem has analogues in the theories developed by Emmy Noether and in categorical formulations influenced by Mac Lane and Eilenberg.

Statement

Let G be a group and let N and H be normal subgroups of G with N ⊆ H. Then the quotient H/N is a normal subgroup of G/N, and there is a natural isomorphism (G/N)/(H/N) ≅ G/H. This statement is typically introduced alongside the role of normal series such as those appearing in the Jordan–Hölder theorem and in analyses of solvable groups studied by Abel and Jordan. The same formulation holds in the contexts of ring theory for two-sided ideals encountered by Dedekind, and in module theory as taught following Noether's approach.

Proof

Let π: G → G/N be the canonical projection used in proofs influenced by the language of Cauchy and Cayley. Since N ⊆ H, the image π(H) equals H/N, and H/N is normal in G/N by virtue of conjugation properties analogous to those employed by Killing in Lie theory. Define φ: G/N → G/H by φ(gN) = gH; φ is well-defined because if gN = g' N then g^{-1}g' ∈ N ⊆ H, so gH = g' H. The map φ is a surjective homomorphism whose kernel is exactly H/N, mirroring classical kernel-image arguments used in proofs of the first isomorphism theorem by figures such as Cayley and Klein. By the first isomorphism theorem applied to φ, (G/N)/(H/N) ≅ G/H, completing the proof.

Examples and applications

A prototypical example arises with finite permutation groups such as the symmetric group S_n and its normal subgroup alternating group A_n: for suitable nested normals one obtains quotients that clarify simplicity properties used by Galois-inspired classifications. In the classification of finite simple groups discussed by contributors including Gorenstein and Thompson, nested normal series and successive quotients are analyzed with this theorem. In linear algebraic settings, for a ring R with ideals I ⊆ J, the theorem yields (R/I)/(J/I) ≅ R/J, a fact used in algebraic geometry developed by Mumford and Grothendieck when manipulating quotient schemes. In representation theory studied by Schur and Weyl, passing to successive quotients clarifies module composition factors as in the Jordan–Hölder theorem framework, and in homological algebra influenced by Henri Cartan and Eilenberg it appears in short exact sequence manipulations.

Variants and generalizations

The theorem generalizes from groups to other algebraic structures: in ring theory it holds for two-sided ideals as in work by Dedekind and Noether; in module theory over a ring R it follows from submodule nesting as in Noetherian module discussions. In the language of category theory pioneered by Mac Lane and Eilenberg, the result is an instance of a statement about quotients by kernels and images, and it extends to Lie algebra ideals studied by Lie and Cartan. Analogues appear in topological group theory and algebraic group theory as developed by Chevalley and Borel, where closed normal subgroups and quotient topologies require additional hypotheses. The theorem is compatible with chain conditions such as Noetherian and Artinian properties emphasized by Noether.

Related results and corollaries

Corollaries include the natural correspondence between subgroups of G containing N and subgroups of G/N, a version of the correspondence theorem used by Jordan and Schreier; applications to composition series and the Jordan–Hölder theorem help classify simple quotients as in work by Lagrange and Galois. The interaction with the first isomorphism theorem and second isomorphism theorem—the trio often attributed to classical algebra texts such as those by Noether and I. N. Herstein—provides a toolkit for decomposition in group theory and ring theory. The theorem also underlies constructions in group cohomology developed by Eilenberg and Steenrod and appears in the study of extensions and factor groups analyzed by Schreier and Schur.

Category:Group theory