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For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.
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For example, for the problem Af, 10 from 25 equals 15, then 6 from 15 equals 9.
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For and homomorphism
For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
For a homomorphism of unital associative R-algebras, we also demand that
For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies
* For any homomorphism f: G → H, f () =.
for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,
For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.
For example, a ring possesses both addition and multiplication, and a homomorphism from the ring to the ring is a function such that
For instance, since a function is bijective if and only if it is both injective and surjective, a module homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.
For instance, the precise definition for a homomorphism f to be iso is not only that it is bijective, and thus has an inverse f < sup >- 1 </ sup >, but also that this inverse is a homomorphism, too.
For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism from R to S is represented by an additive functor from R to S, and conversely.
For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups.
For an algebra, given a congruence E on, the algebra is called the quotient algebra ( or factor algebra ) of modulo E. There is a natural homomorphism from to mapping every element to its equivalence class.
For a compact topological group, G, there exists a C *- algebra homomorphism Δ: C ( G ) → C ( G ) ⊗ C ( G ) ( where C ( G ) ⊗ C ( G ) is the C *- algebra tensor product-the completion of the algebraic tensor product of C ( G ) and C ( G )), such that Δ ( f )( x, y )
For every n there is a surjective ring homomorphism ρ < sub > n </ sub > from the analoguous ring R < sup > S < sub > n + 1 </ sub ></ sup > with one more indeterminate onto R < sup > S < sub > n </ sub ></ sup >, defined by setting the last indeterminate X < sub > n + 1 </ sub > to 0.
* For every index set I, the addition map M < sup >( I )</ sup > → M can be extended to a module homomorphism M < sup > I </ sup > → M ( here M < sup >( I )</ sup > denotes the direct sum of copies of M, one for each element of I ; M < sup > I </ sup > denotes the product of copies of M, one for each element of I ).
For the smallest positive integer such that there exists a computable injective group homomorphism from the subgroup of of order to.
There is a unique ring homomorphism φ from Z to Z / nZ that maps α to m. For simplicity, we'll assume that Z is a unique factorization domain ; the algorithm can be modified to work when it isn't, but then there are some additional complications.
For each such pair, we can apply the ring homomorphism φ to the factorization of a + bα, and we can apply the canonical ring homomorphism from Z to Z / nZ to the factorization of a + bm.
For any space X and positive integer k there exists a group homomorphism
For any pair of spaces ( X, A ) and integer k > 1 there exists a homomorphism
For any triad of spaces ( X ; A, B ) ( i. e. space X and subspaces A, B ) and integer k > 2 there exists a homomorphism

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