Question

How many different homomorphisms o : Z/5ZZ/20Z are there?

Answer 1

2

Explanation:

We are given that

`O:Z/5Z\rightarrow Z/20Z`

We have to find number of different homomorphisms

We know that

Z/5Z is isomorphic to
`Z_5` and
`Z/20Z` isomorphic to
`Z_(20)`

We know that

If
`O:Z_n\rightarrow Z_m`

Number of homomorphism=gcd(m,n)

Gcd(5,10)=2

5 does not divide by 20 therefore, there are no onto homomorphisms.

There are only two homomorphisms.