## [Homomorphism](/homomorphism-0) Definition A map between algebraic structures that preserves the operations and relations specified by those structures (e.g., f(ab)=f(a)f(b) for group homomorphisms). ## Principle Principle A homomorphism respects the defining operations of a structure so that algebraic identities and relations in the source are mapped to valid identities in the target. ## Demonstration Demonstration Group homomorphism: φ: G → H with φ(ab)=φ(a)φ(b); kernel(φ) is a normal subgroup and image(φ) is a subgroup of H. Ring homomorphism preserves addition, multiplication, and multiplicative identity when required. ## Misapplication Misapplication Assuming a homomorphism is injective or surjective without proof; assuming it preserves auxiliary structure not specified (e.g., topology or order) or that kernels always split. ## Consequence Consequence Kernels and images yield substructures and allow formation of quotient structures; homomorphisms organize objects into exact sequences and permit transfer of algebraic information. ## Reversal Reversal An arbitrary function ignoring the operations is not a homomorphism; a bijection that fails to preserve operations is not one either. ## Boundary Boundary Depends on the specified operations and relations; what counts as 'preserving' must be made explicit (groups vs rings vs modules). In categorical language a homomorphism is a particular kind of morphism defined by the algebraic category. ## Semantic Tension Semantic Tension 'Structure-preserving' is broad — homomorphism emphasizes preservation of algebraic operations, whereas isomorphism requires invertibility; homomorphism versus embedding (injectivity) is a common point of confusion. ## Synthesis Synthesis A homomorphism is the map that transmits algebraic structure from one object to another by converting operations and relations of the source into valid operations and relations of the target.