How many homomorphisms are there from Z20 onto Z10 How many are there to Z10?

4 homomorphisms
To have an image of Z10, φ(1) must generate Z10. Hence, φ(1) is either 1, 3, 7, or 9. So there are 4 homomorphisms onto Z10.

Is there always a homomorphism between two groups?

Group homomorphism always exist between two groups.

How do you find the order of a group Homomorphism?

Theorem

1. Let G and H be groups whose identities are eG and eH respectively.
2. Let ϕ:G→H be a homomorphism.
3. Let g∈G be of finite order.
4. where ∖ denotes divisibility.
5. Let ϕ:G→H be a homomorphism.
6. Let |g|=n,|ϕ(g)|=m.
7. It follows from Element to Power of Multiple of Order is Identity that m∖n.

How many homomorphisms are there from Z20 onto Z8 how many are there to Z8?

There is no homomorpphism from Z20 onto Z8. If φ : Z20 → Z8 is a homomorphism then the order of φ(1) divides gcd(8,20) = 4 so φ(1) is in a unique subgroup of order 4 which is 2Z8. Thus possible homomorphisms are of the form x → 2i · x where i = 0,1,2,3.

Can there be a homomorphism from Z4 Z4 onto Z8?

– Can there be a homomorphism from Z4 ⊕ Z4 onto Z8? No. If f : Z4 ⊕ Z4 −→ Z8 is an onto homomorphism, then there must be an element (a, b) ∈ Z4 ⊕ Z4 such that |f(a, b)| = 8. This is impossible since |(a, b)| is at most 4, and |f(a, b)| must divide |(a, b)|.

How many homomorphism are there of Z onto Z?

Because all homomorphisms must take identities to identities, there do not exist any more homomorphisms from Z to Z. Clearly, the identity map is the only surjective mapping. Thus there exists only one homomorphism from Z to Z which is onto.

What is homomorphism and isomorphism in group theory?

Isomorphism. A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.

Which of the following is not a group wrt addition?

The set of odd integers under addition is not a group.

Is there a group of order 2?

There is, up to isomorphism, a unique group of order 2, namely cyclic group:Z2.

Can there be a homomorphism from Z4 Z4 onto Z8 can there be a homomorphism from z16 onto Z2 Z2?

Can there be a homomorphism from Z8 ⊕ Z2 onto Z4 ⊕ Z4 give reasons for your answer?

Because |Z8 ⊕ Z2| = 16 = |Z4 ⊕ Z4|, if φ is onto, then it is an isomorphism. But Z8 ⊕ Z2 has an element of order 8 ((1,0)), and all elements of Z4 ⊕ Z4 have order at most 4. Therefore they are not isomorphic. For any homomorphism φ : Z4 ⊕ Z4 → Z8, |φ(a)|≤|a| ≤ 4 because any element in Z4 ⊕ Z4 has order at most 4.

What is a group homomorphism?

Group Homomorphisms. Definitions and Examples Definition (Group Homomorphism). A homomorphism from a group G to a group G is a mapping : G ! G that preserves the group operation: (ab) = (a)(b) for all a,b 2 G. Definition (Kernal of a Homomorphism).

What is the kernel of a homomorphism?

The kernel of a homomorphism : G ! G is the set Ker = {x 2 G|(x) = e} Example. (1) Every isomorphism is a homomorphism with Ker = {e}. (2) Let G = Z under addition and G = {1,1} under multiplication.

What is the difference between monomorphism and isomorphism?

A one to one (injective) homomorphism is a monomorphism. An onto (surjective) homomorphism is an epimorphism. A one to one and onto (bijective) homomorphism is an isomorphism. If there is an isomorphism from G to H, we say that G and H are isomorphic, denoted G ∼= H.