an induced homomorphism. A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often...

of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves...

¯ c ∗ ( X , A ; G ) {\displaystyle {\bar {H}}_{c}^{*}(X,A;G)} . Induced homomorphism of this cohomology is defined as the Alexander cohomology theory...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function...

be different.) On the other hand, a module homomorphism M → K is a pure embedding if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K...

{\displaystyle [k]} is a ring homomorphism and is called the de Rham homomorphism. Finally, the theorem says that the induced homomorphism [ k ] {\displaystyle...

quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined). If A is...

_{0}(Y)} is bijective, and for every point x in X and every n ≥ 1, the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) {\displaystyle f_{*}\colon...

properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the...

definitions. A homomorphism of topological groups means a continuous group homomorphism G → H. Topological groups, together with their homomorphisms, form a...

the (usual) Gysin homomorphism induced by the zero-section embedding X ′ ↪ N {\displaystyle X'\hookrightarrow N} . The homomorphism i! encodes intersection...

substructures are precisely the induced subgraphs. However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding...

Then, for a homomorphism f : G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to...

{Gray}\longrightarrow }~\operatorname {coker} c} where d is a homomorphism, known as the connecting homomorphism. Furthermore, if the morphism f is a monomorphism...

functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological...

characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism ( R , m ) ↪ ( S , n ) {\displaystyle...

of bits Ω = 2 Z , {\displaystyle \Omega =2^{\mathbb {Z} },} the induced homomorphism is the Baker's map. Consider now the space of functions in y {\displaystyle...

{\displaystyle f\colon X\to Y} and a point x in X, consider for any n ≥ 1 the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) , {\displaystyle f_{*}\colon...

exists a group homomorphism h ∗ : π n ( X ) → H n ( X ) , {\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),} called the Hurewicz homomorphism, from the n-th...

and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as π ( f ) {\displaystyle \pi (f)} or, more...

∗ ( X ) {\displaystyle f^{*}\colon H^{*}(Y)\to H^{*}(X)} is the induced homomorphism in cohomology, then f ∗ ( α ⌣ β ) = f ∗ ( α ) ⌣ f ∗ ( β ) , {\displaystyle...

isometries of (T, d), with the quotient space T/π1(Γ) = Γ. Since the induced homomorphism f# is an isomorphism between Fn = π1(Rn) and π1(Γ), we also obtain...

isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over...

group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object. Homomorphism groups...

continuous homomorphism between real Lie groups turns out to be (real) analytic. The composition of two Lie homomorphisms is again a homomorphism, and the...

of bits Ω = 2 Z , {\displaystyle \Omega =2^{\mathbb {Z} },} the induced homomorphism is the Baker's map. The dyadic sequence is then just the sequence...

covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering...

continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but...

{\textstyle d\neq q} with d ∣ q {\textstyle d\mid q} such that the induced homomorphism χ ~ : ( Z / q Z ) × → C ∗ {\textstyle {\widetilde {\chi }}\colon...

B ) {\displaystyle h\colon (X,A)\rightarrow (Y,B)} , then their induced homomorphisms are the same. Excision: If ( X , A ) {\displaystyle (X,A)} is a...