I am not sure if the following is correct, could someone confirm
O : C* ---> C*
z ----> zz
O (z1Z2) = z1z2 x z1z2 = z1z1 x z2z2 = O (z1) O (z2)
Thus O is a homomorphism
C* is an abelian group under multiplication, for any abelian group $\displaystyle G: g\rightarrow g^2$ is a homomorphism:
$\displaystyle (gh)^2 = (gh)(gh) = g(hg)h = g(gh)h = (gg)(hh) = g^2h^2$.
the fact that C* is indeed abelian under multiplication is worth verifying once
Sorry I do mean C as in complex numbers. Thanks.
Can anyone tell me how to get the advanced features.
One other thing, is the following correct?
O : C ----> C
z ---> 3/z
O(z1 x z2) = 3/z1 x z2 = 3/z1 x 3/z2 = O(z1) O(z2)
thus O is a homomorphism.
The advanced features are LaTeX. There is a LaTeX tutorial somewhere on this site...alternatively, search for `LaTeX commands' on google (but forget all the stuff about dollar signs and \[s 'cause they aren't used here).
To put it in LaTeX, put [tex] at the start of what you are wanting to make into LaTeX and [/ tex] at the end (without the space). For example,
[tex]\phi: \mathbb{C}\rightarrow \mathbb{C}, z\mapsto 3/z[/ tex] will give you $\displaystyle \phi: \mathbb{C}\rightarrow \mathbb{C}, z\mapsto 3/z$, and if you want to be ultra fancy, do a \frac{3}{z} to make the 3/z into a fraction, $\displaystyle \frac{3}{z}$. (the \mathbb{...} puts the thing in the brackets into blackboard bold, and everything is case-sensitive, so \rightarrow is different from \Rightarrow...)
Just want make sure I get this right. Are you saying
C is the group of all complex numbers under addition.
H : C ---> C
z -----> z + i
The function H is not a homomorphism.
For example
H(z1+z2) = 0
but H (z1) H(z2) = 0+0i
Hence H does not have the homomorphism property.
I have one question dont these both add up to 0.