How many Homomorphisms are there from Z onto Z?

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?

$\\begingroup$@Kuttus: 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