A few examples of group homomorphism

1. May 25, 2010

zcahana

1. The problem statement, all variables and given/known data

The exercise is to find examples of various homomorphisms from/to various groups.

Those I'm having problems with are:

a. f: (Q,+) --> (Q^+,*) which is onto.
b. f: U20 --> Z64 which is 1-to-1.
c. f: Z30 --> S10 which is 1-to-1.

2. Relevant equations

3. The attempt at a solution

for a. , the only homomorphism i could think which send 0 to 1 is f(x) a^x,
apparently, f sends many elements of (Q,+) to R instead of (Q^+,*).

for b. , the only insight i have is that U20 has 8 elements, and Z64 has 8*8 elements ... not to deep, i know ... but maybe is?

for c. , no insights at all.

I would appreciate any clues given.
Thanks,
Zvi.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 25, 2010

Martin Rattigan

To get b. and c. out of the way first (then we can come back to a. which is more interesting).

b. isn't possible if by $U_{20}$ you mean the group of residues relatively prime to 20 under multiplication. This is because, with that meaning, $U_{20}\cong Z_4\times Z_2$ which is not cyclic (prove both, or at any rate the latter!), whereas any subgroup of a cyclic group ($Z_{64}$ in this case) is cyclic (also prove!).

c. Does (12) commute with (345)? What then is the order of (12)(345). Then find an element of order 30 in $S_{10}$.

Last edited: May 25, 2010
3. May 25, 2010

Martin Rattigan

a. is also not possible. Look at an element $x\in (\mathbb{Q},+)$ such that $f:x\mapsto 2\in(\mathbb{Q}^+,\times)$ (which must exist). What can you say about $x/2\text{?}$

4. May 26, 2010

zcahana

Ok,

For a. , I concluded that if f(x) = 2 then f(x/2) = sqrt(2) , with contradiction to that f(x/2) is (positive) rational.

For b. , proved.

For c. , I marked s = (1,2)(3,4,5)(6,7,8,9,10) , o(s) = 30
and for any x in Z30, f(x) = s^x such that IM(f) = <s> , and f(x) is well defined because it's not dependant of the number one chooses to represent the residue x.

Thanks for your help, Martin ;)