Table of Contents

- 1 How many Homomorphisms are there from Z onto Z?
- 2 How many group Homomorphisms are there from z8 into Z10?
- 3 How many group homomorphisms are there from?
- 4 How do you identify group Homomorphisms?
- 5 How many group homomorphisms are there from z m into z n?
- 6 How do you find the number of ring homomorphisms?

## How many Homomorphisms are there from 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.

### How many group Homomorphisms are there from z8 into Z10?

So there are 4 homomorphisms onto Z10. Now, let’s examine homomorphisms to Z10. Then φ(1) must have an order that divides 10 and that divides 20.

#### How many group homomorphisms are there from?

So there are four homomorphisms, each determined by choosing the common image of a,b.

**Can there be a Homomorphism from Z4 Z4 onto Z8 can there be a Homomorphism from z16 onto z2 z2 explain your answers?**

– 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)|.

**Are all Isomorphisms Homomorphisms?**

Therefore, all the three homomorphisms are isomorphisms. A map f:F→G is one-to-one and onto if and only if it has an inverse map, i. e. a map g:G→F such that g(f(x))=x for all x∈F and f(g(y))=y for all y∈G.

## How do you identify group Homomorphisms?

If h : G → H and k : H → K are group homomorphisms, then so is k ∘ h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.

### How many group homomorphisms are there from z m into z n?

The number of group homomorphisms from Z m into Z n is gcd ( m, n). The number of ring homomorphisms from Z m into Z n is 2 ω ( n) − ω ( n / gcd ( m, n)) where ω ( a) denotes the number of distinct prime divisors of the integer a.

#### How do you find the number of ring homomorphisms?

A ring homomorphism f: Z m → Z n is uniquely determined by the conditions: m f (1) = 0 and f (1) 2 = f (1). In order to find out how many ring homomorphisms there are we have to count the number of elements of the set { e ∈ Z n: e 2 = e, m e = 0 }.

**How do you find the Order of a homomorphism with cyclic domain?**

A group homomorphism with cyclic domain is completely determined by the image of a generator. If f: G → H is a homomorphism, and x ∈ G, then the order of f(x) must be a divisor of the order of x. Since the only divisors of 3 are 1 and 3, the answer is one plus the number of elements of Z6 of order 3…

**Do ring homomorphisms have to preserve multiplicative identities?**

$\\[email protected]: Even if you don’t require ring homomorphisms to preserve multiplicative identities, you cannot send 1 to any element of $Z_{28}$. Homomorphisms send 0 to 0, and must preserve addition…$\\endgroup$ – Zev Chonoles Dec 21 ’12 at 6:05 1