## Group theory question

Forum for the GRE subject test in mathematics.
mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Group theory question

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

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

### Re: Group theory question

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

mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Re: Group theory question

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

Can you explain how you reached from 6+6+6 = 2 ?

I might be missing some basic fundamentals here.

Thanks

congvan
Posts: 9
Joined: Sat Mar 06, 2010 4:44 am

### Re: Group theory question

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

mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Re: Group theory question

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

why did you use the modulus operation ?

congvan
Posts: 9
Joined: Sat Mar 06, 2010 4:44 am

### 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.

mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Re: Group theory question

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.
So any Zn Group with '+' as binary operation contains following values :

Int mod n ?

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

origin415
Posts: 61
Joined: Fri Oct 23, 2009 11:42 pm

### Re: Group theory question

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 ?
Yes. Its the cyclic group.

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.