Group homomorphism

Group homomorphism
Name	Group homomorphism
Field	Mathematics
Subfield	Évariste Galois, Niels Henrik Abel, Émile Picard
Introduced	19th century
Related	Abstract algebra, Ring homomorphism, Module (mathematics), Category theory

Group homomorphism

A group homomorphism is a structure-preserving map between two algebraic structures called groups that respects the binary operation. It originated in the development of Évariste Galois's work on permutation groups, was formalized through the contributions of Niels Henrik Abel and Camille Jordan, and became central to Emmy Noether's structural approach to algebra. Homomorphisms connect concrete examples like symmetry groups arising in Isaac Newton's work on mechanics and abstract classifications used by David Hilbert and Hermann Weyl.

Definition

Given two groups G and H with operations written multiplicatively, a map f: G → H is a group homomorphism if for all a, b in G one has f(ab) = f(a)f(b). This condition appears in the foundational texts of Augustin-Louis Cauchy and was systematized in lectures by Arthur Cayley and Felix Klein. The identity element requirement f(e_G) = e_H and the inversion-preserving property f(a^{-1}) = f(a)^{-1} follow from the defining equation; these consequences are noted in expositions by Issai Schur and Otto Schreier. Homomorphisms are morphisms in the category of groups, a viewpoint championed by Saunders Mac Lane and Samuel Eilenberg.

Examples

Standard examples include the determinant map det: GL_n(ℝ) → ℝ^× studied by Carl Friedrich Gauss and Adrien-Marie Legendre, the sign homomorphism sign: S_n → {±1} prominent in the work of Augustin-Louis Cauchy and Arthur Cayley, and the exponential map from (ℝ, +) to (ℝ_{>0}, ×) used in analyses by Leonhard Euler and Joseph Fourier. Quotient projections G → G/N for a normal subgroup N were central to Camille Jordan's composition series investigations and recur in George Boole's algebraic logic. Inclusion maps of subgroups, trivial homomorphisms, and conjugation maps g ↦ hgh^{-1} studied by William Rowan Hamilton and Sophus Lie are routine examples. Covering space monodromy representations connect to work of Henri Poincaré and Ludwig Bieberbach.

Kernel and Image

The kernel ker(f) = {g ∈ G | f(g) = e_H} and the image im(f) = f(G) are central invariants of a homomorphism; these notions feature in the writings of Évariste Galois and Richard Dedekind. The kernel is a normal subgroup of G and the image is a subgroup of H, a fact used in classification problems tackled by Issai Schur and Philip Hall. The First Isomorphism Theorem, formulated by Noetherian school mathematicians such as Emmy Noether and later streamlined by Oscar Zariski, states G/ker(f) ≅ im(f), a result that unifies many constructions across studies by Hermann Weyl and Claude Shannon's algebraic analogues. Kernels detect injectivity: f is injective iff ker(f) is trivial, a criterion employed in proofs by Sophus Lie and Élie Cartan.

Properties and Theorems

Homomorphisms compose: if f: G → H and g: H → K then g∘f: G → K is a homomorphism, reflecting categorical composition emphasized by Saunders Mac Lane. Images and kernels interact through the lattice isomorphism theorems developed by Emmy Noether and Bartel Leendert van der Waerden. The Correspondence Theorem relates subgroups of G containing ker(f) to subgroups of im(f), an idea used in William Burnside's group enumeration methods and Burnside's lemma applications. Sylow theorems, proved by Ludvig Sylow, constrain possible homomorphic images of finite groups; representation theory as developed by Ferdinand Frobenius links homomorphisms to linear actions on vector spaces in the style of Richard Brauer and Issai Schur. Universal properties characterize free groups and direct products via unique homomorphisms, an approach central to Nicholas Bourbaki's structural program.

Homomorphisms and Group Actions

Every group action of G on a set X yields a homomorphism from G to the symmetric group Sym(X), an observation dating back to Cayley's theorem. Conversely, homomorphisms into Sym(X) describe permutation representations used by Évariste Galois in solvability criteria and by William Rowan Hamilton in quaternionic symmetries. Actions on geometric objects studied by Felix Klein and Henri Poincaré produce homomorphisms into isometry groups like O(n) and SO(n), linking classical geometry with algebraic structure. Monodromy homomorphisms in algebraic topology, developed by Henri Poincaré and Lefschetz, associate covering transformations to fundamental groups, while holonomy maps in differential geometry are treated in the works of Élie Cartan and Marcel Berger.

Classification and Applications

Classification of homomorphisms between specific groups underpins major theories: classification of finite simple groups by organizations of mathematicians culminating in the work of Daniel Gorenstein and Robert Guralnick informs possible surjective homomorphisms from finite groups; character theory by Ferdinand Frobenius and Issai Schur describes homomorphisms into GL_n(ℂ). In number theory, Galois representations—homomorphisms from Galois groups first studied by Évariste Galois and furthered by Andrew Wiles and Jean-Pierre Serre—connect to modular forms and Taniyama–Shimura-type conjectures. Crystallography uses homomorphisms into space groups, a field shaped by Ludwig Bieberbach and industrial research institutions. Computational group theory, advanced at centers like Bonn University and University of Warwick, implements homomorphism tests and quotient constructions for algorithmic classification and applications in coding theory and cryptography influenced by Claude Shannon and Whitfield Diffie.

Category:Group theory