## Group theory question

### Group theory question

Let f:(Z4,+) --> (Z8,+) be a homomorphism such that f(1) = 6. Find f(3) ?

### Re: Group theory question

f(3) = f(1+1+1) = f(1)+f(1)+f(1) = 6+6+6 = 2

### Re: Group theory question

Thanks lime.lime wrote:f(3) = f(1+1+1) = f(1)+f(1)+f(1) = 6+6+6 = 2

Can you explain how you reached from

**6+6+6 = 2**?

I might be missing some basic fundamentals here.

Thanks

### Re: Group theory question

Since 6 + 6 + 6 = 18, 18 in Z8 is 18 mod 8 = 2.

### Re: Group theory question

Thanks congvan.congvan wrote:Since 6 + 6 + 6 = 18, 18 in Z8 is 18 mod 8 = 2.

why did you use the modulus operation ?

### Re: Group theory question

Since Z8 only contains the integer mod 8, i.e. {0,1,2,3,4,5,6,7}. Generally Zn only contains the integer mod n.

### Re: Group theory question

So any Zn Group with '+' as binary operation contains following values :congvan wrote:Since Z8 only contains the integer mod 8, i.e. {0,1,2,3,4,5,6,7}. Generally Zn only contains the integer mod n.

**Int mod n**?

Secondly what about Zn Group with 'x' as binary operation ?

### Re: Group theory question

Yes. Its the cyclic group.mathQ wrote: So any Zn Group with '+' as binary operation contains following values :

Int mod n?

Secondly what about Zn Group with 'x' as binary operation ?

Zn with multiplication will only be a group if n is prime, otherwise not all elements will have inverses. If it is a group, then the operation is just taking two numbers and multiplying them mod n just as before.