Homomorphism of cyclic group

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• Kernel of Homomorphism

  : is a group homomorphism. is onto. Since is a cyclic group, then is also a cyclic group generated by . From the condition, has order 4. It has only two structures: or . is a cyclic group, is not. Thus .

• Continuous Maps, Homomorphisms and Cyclic Groups

  How exactly does "there is a unique integer d such that..." follow from "H~_n (S^n) is an infinite cyclic group"? Call H~_n(S^n) G for short. Take a generator g of the group G, so G = {1, g, g^2, g^3 ,...., g^{-1}, g^{-2},... }.

• Homomorphisms

  Since H is a cyclic group, animage of the homomorphism would have an order that was a divisor of the order of H. 3. G and H have relatively prime orders, so they don't share any divisors other than 1.

• Homomorphism Subgroup Proofs

  Prove that the only homomorphism from G to H is the trivial homomorphism. (b) Give an example of a group G of size 77 and a group K of size 56 and a nontrivial homomorphism from G to IC.

• Group homomorphisms

  Under any group homomorphism µ : Z4 -> Z10, the order of µ ([1]4) must be a divisor of 4 and of 10, so the only possibilities are 1 and 2.

• Homomorphisms and Mapping Prime Elements

  Therefore it is impossible to have any homomorphism as stated. ### QED Notes:- (*1) For ANY element x in a cyclic group C with C elements, (1.1) x^c = e (the identity element).

• Kernel of Phi and Homomorphism

  359985 Kernel of Phi and Homomorphism Let phi is a homomorphism from Z30 onto a group order 3. Determine the kernel of phi. Find all generators of the kernel of phi.

• Ideals : Cyclic Module in a Commutative Ring

  We can see f is an additive homomorphism of groups.

• Group Theory : Homomorphism, Subgroups, Abelian Groups and Group Order

  Let G be an abelian group of order 16. Write down all direct products of cyclic groups that could be isomorphic to G. If G has elements x and y, each of order 4, with x2 6= y2, can you determine G (up to isomorphism)?