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Homomorphism — Grokipedia Fact-checked by Grok 3 months ago Homomorphism Ara Eve Leo Sal 1x In mathematics , particularly within abstract algebra , a homomorphism is a function between two algebraic structures of the same type—such as groups, rings, or vector spaces—that preserves the operations and relations defining those structures. [1] For instance, in the case of groups G G G and H H H , a homomorphism ϕ : G → H \phi: G \to H ϕ : G → H satisfies ϕ ( g 1 g 2 ) = ϕ ( g 1 ) ϕ ( g 2 ) \phi(g_1 g_2) = \phi(g_1) \phi(g_2) ϕ ( g 1 ​ g 2 ​ ) = ϕ ( g 1 ​ ) ϕ ( g 2 ​ ) for all g 1 , g 2 ∈ G g_1, g_2 \in G g 1 ​ , g 2 ​ ∈ G , ensuring compatibility with the group multiplication . [2] Similarly, for rings R R R and S S S , it preserves both addition and multiplication : ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) \phi(a + b) = \phi(a) + \phi(b) ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) and ϕ ( a b ) = ϕ ( a ) ϕ ( b ) \phi(ab) = \phi(a) \phi(b) ϕ ( ab ) = ϕ ( a ) ϕ ( b ) for all a , b ∈ R a, b \in R a , b ∈ R . [3] Key properties of homomorphisms include their images and kernels, which provide insights into the structural relationships between the domain and codomain . The image of a homomorphism ϕ : G → H \phi: G \to H ϕ : G → H is the subset ϕ ( G ) ⊆ H \phi(G) \subseteq H ϕ ( G ) ⊆ H , which forms a subgroup of H H H . [2] The kernel is the preimage of the identity element in H H H , defined as ker ⁡ ( ϕ ) = { g ∈ G ∣ ϕ ( g ) = e H } \ker(\phi) = \{ g \in G \mid \phi(g) = e_H \} ker ( ϕ ) = { g ∈ G ∣ ϕ ( g ) = e H ​ } , and it is always a normal subgroup of G G G . [2] These elements enable the formation of quotient structures; for example, the first isomorphism theorem states that G / ker ⁡ ( ϕ ) ≅ im ⁡ ( ϕ ) G / \ker(\phi) \cong \operatorname{im}(\phi) G / ker ( ϕ ) ≅ im ( ϕ ) , linking the original group to a simpler isomorphic copy. [4] Homomorphisms are foundational to understanding algebraic similarities and classifications, as they reveal how one structure can be embedded into or projected onto another. A bijective homomorphism, called an isomorphism , establishes that two structures are essentially identical up to relabeling, while injective ones (monomorphisms) allow embeddings as substructures. [1] Examples abound in applications: the exponential map x ↦ e x x \mapsto e^x x ↦ e x is a homomorphism from ( R , + ) (\mathbb{R}, +) ( R , + ) to ( R + , × ) (\mathbb{R}^+, \times) ( R + , × ) , and the determinant function is a homomorphism from the general linear group G L n ( R ) GL_n(\mathbb{R}) G L n ​ ( R ) to the multiplicative group R × \mathbb{R}^\times R × . [2] Beyond groups and rings, homomorphisms extend to modules, fields, and categories, underpinning theorems in Galois theory and representation theory. [1] Core Concepts Definition In category theory , a homomorphism is synonymous with a morphism , which is an arrow $ f: A \to B $ between objects $ A $ and $ B $ in a category. Categories consist of objects (such as sets or algebraic structures) and morphisms (maps between them) with associative composition and identity morphisms obeying the category axioms. [5] [6] In algebra , a homomorphism is a function $ f: S \to T $ between two algebraic structures $ S $ and $ T $ of the same type that preserves their operations; for instance, in groups $ (S, \cdot_S) $ and $ (T, \cdot_T) $, it satisfies $ f(a \cdot_S b) = f(a) \cdot_T f(b) $ for all $ a, b \in S $, and similarly for rings (preserving addition and multiplication ) or vector spaces (preserving addition and scalar multiplication). [7] [5] Note that for rings, some definitions require homomorphisms to preserve the multiplicative identity (unital ring homomorphisms), while others do not. For algebraic structures with compatible identity elements, such as groups, homomorphisms map the identity to the identity; similar conventions apply to unital rings and vector spaces. Standard notation employs $ f $ or $ \phi $ for these functions, with the preservation conditions ensuring the map respects the underlying algebraic relations without altering the intrinsic operations. [5] These definitions presuppose familiarity with categories as collections of objects and morphisms, and algebraic structures like groups, rings, or vector spaces equipped with compatible operations. [6] A bijective homomorphism whose inverse is also a homomorphism is termed an isomorphism , establishing structural equivalence between objects. [8] Properties Homomorphisms possess fundamental preservation properties that maintain the structural integrity of the source and target objects. In the context of groups, a homomorphism ϕ : G → H \phi: G \to H ϕ : G → H maps the identity element of G G G to the identity element of H H H , i.e., ϕ ( e G ) = e H \phi(e_G) = e_H ϕ ( e G ​ ) = e H ​ . [9] It also preserves inverses, satisfying ϕ ( g − 1 ) = ϕ ( g ) − 1 \phi(g^{-1}) = \phi(g)^{-1} ϕ ( g − 1 ) = ϕ ( g ) − 1 for all g ∈ G g \in G g ∈ G . [10] These properties extend to other algebraic structures, where homomorphisms preserve the defining operations and relations; for instance, in partially ordered sets (posets), a homomorphism f : P → Q f: P \to Q f : P → Q is order-preserving, meaning if a ≤ P b a \leq_P b a ≤ P ​ b then f ( a ) ≤ Q f ( b ) f(a) \leq_Q f(b) f ( a ) ≤ Q ​ f ( b ) . [11] A key universal property of homomorphisms is their closure under composition. If ϕ : G → H \phi: G \to H ϕ : G → H and ψ : H → K \psi: H \to K ψ : H → K are homomorphisms between groups, then the composite map ψ ∘ ϕ : G → K \psi \circ \phi: G \to K ψ ∘ ϕ : G → K is also a homomorphism. [2] This composition property underpins the formation of categories, where objects are the algebraic structures and morphisms are the homomorphisms. [12] The first isomorphism theorem provides a structural relation between the kernel and image of a homomorphism (see Structural Elements). For a group homomorphism ϕ : G → H \phi: G \to H ϕ : G → H , there exists a natural isomorphism G / ker ⁡ ( ϕ ) ≅ im ⁡ ( ϕ ) G / \ker(\phi) \cong \operatorname{im}(\phi) G / ker ( ϕ ) ≅ im ( ϕ ) , where ker ⁡ ( ϕ ) \ker(\phi) ker ( ϕ ) is the preimage of the identity in H H H . [13] In topological contexts, such as topological groups, homomorphisms are often continuous, thereby preserving limits and the topological structure. [14] Examples Algebraic Examples In group theory, a concrete example of a homomorphism is the projection map from the direct product group Z × Z \mathbb{Z} \times \mathbb{Z} Z × Z to Z \mathbb{Z} Z , defined by f ( m , n ) = m f(m, n) = m f ( m , n ) = m . This map preserves the group operation of addition: for any ( m , n ) , ( p , q ) ∈ Z × Z (m, n), (p, q) \in \mathbb{Z} \times \mathbb{Z} ( m , n ) , ( p , q ) ∈ Z × Z , f ( ( m , n ) + ( p , q ) ) = f ( m + p , n + q ) = m + p = f ( m , n ) + f ( p , q ) f((m, n) + (p, q)) = f(m + p, n + q) = m + p = f(m, n) + f(p, q) f (( m , n ) + ( p , q )) = f ( m + p , n + q ) = m + p = f ( m , n ) + f ( p , q ) . [15] In ring theory , the inclusion map from the ring of integers Z \mathbb{Z} Z to the field of rational numbers Q \mathbb{Q} Q , given by f ( k ) = k f(k) = k f ( k ) = k for all k ∈ Z k \in \mathbb{Z} k ∈ Z , is a ring homomorphism . It preserves both addition and multiplication: f ( a + b ) = a + b = f ( a ) + f ( b ) f(a + b) = a + b = f(a) + f(b) f ( a + b ) = a + b = f ( a ) + f ( b ) and f ( a ⋅ b ) = a ⋅ b = f ( a ) ⋅ f ( b ) f(a \cdot b) = a \cdot b = f(a) \cdot f(b) f ( a ⋅ b ) = a ⋅ b = f ( a ) ⋅ f ( b ) , and it maps the multiplicative identity 1 ∈ Z 1 \in \mathbb{Z} 1 ∈ Z to 1 ∈ Q 1 \in \mathbb{Q} 1 ∈ Q . [16] In the context of vector spaces over the real numbers, any linear transformation T : R n → R m T: \mathbb{R}^n \to \mathbb{R}^m T : R n → R m serves as a homomorphism, preserving vector addition and scalar multiplication : T ( u + v ) = T ( u ) + T ( v ) T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) T ( u + v ) = T ( u ) + T ( v ) and T ( c u ) = c T ( u ) T(c \mathbf{u}) = c T(\mathbf{u}) T ( c u ) = c T ( u ) for u , v ∈ R n \mathbf{u}, \mathbf{v} \in \mathbb{R}^n u , v ∈ R n and c ∈ R c \in \mathbb{R} c ∈ R . Such transformations admit a matrix representation : relative to standard bases, T ( x ) = A x T(\mathbf{x}) = A \mathbf{x} T ( x ) = A x where A A A is an m × n m \times n m × n matrix whose columns are the images of the standard basis vectors of R n \mathbb{R}^n R n . [17] Categorical Examples In category theory , homomorphisms are precisely the morphisms of a category, which abstract the notion of structure-preserving maps across various mathematical domains. A foundational example occurs in the category Set , where the objects are sets and the morphisms—termed homomorphisms—are arbitrary functions between sets. Composition of these homomorphisms corresponds to the standard function composition , ensuring that the diagram of sets and functions satisfies the categorical axioms of associativity and identity preservation. [18] Another illustrative case arises in the category of partially ordered sets, often denoted Poset , where objects are posets and homomorphisms are order-preserving maps. Such a map f : ( P , ≤ P ) → ( Q , ≤ Q ) f: (P, \leq_P) \to (Q, \leq_Q) f : ( P , ≤ P ​ ) → ( Q , ≤ Q ​ ) satisfies x ≤ P y x \leq_P y x ≤ P ​ y implies f ( x ) ≤ Q f ( y ) f(x) \leq_Q f(y) f ( x ) ≤ Q ​ f ( y ) for all x , y ∈ P x, y \in P x , y ∈ P , thereby maintaining the partial order structure under the mapping. These homomorphisms form the arrows of Poset , with composition defined pointwise as in Set , highlighting how categorical homomorphisms generalize relational preservation beyond algebraic operations. [19] Homomorphisms can also be induced by functors, which are structure-preserving maps between categories themselves. A functor F : C → D F: \mathcal{C} \to \mathcal{D} F : C → D sends objects of C \mathcal{C} C to objects of D \mathcal{D} D and morphisms (homomorphisms) of C \mathcal{C} C to morphisms of D \mathcal{D} D , preserving composition