# Thread: Inverse of a homomorphic function

1. ## Inverse of a homomorphic function

Let f(A,)-->(B,) be homomorphic. Prove that if f1 is a function then it is homomorphic
Inorder for
f1 to be a function then f must be a bijection

2. ## Re: Inverse of a homomorphic function

First, in order for $\displaystyle f^{-1}$ to exist, f must be "onto". That is, for every y in B there must be x in A such that f(x)= y. Otherwise we could not define $\displaystyle f^{-1}(y)$. If f were not "one-to-one", there would be some y in B such that for both $\displaystyle x_1$ and $\displaystyle x_2$ in A, $\displaystyle f(x_1)= y= f(x_2)$. In that case we would have both $\displaystyle f^{-1}(y)= x_1$ and $\displaystyle f^{-1}(y)= x_2$, contradicting the fact that $\displaystyle f^{-1}$ is a function.

f being a homomorphism means $\displaystyle f(x_1*x_2)= f(x_1).f(x_2)$ for all $\displaystyle x_1$ and $\displaystyle x_2$ in A. To show that $\displaystyle f^{-1}$ is a homorphism, you must use that to show that $\displaystyle f^{-1}(y_1.y_2)= f^{-1}(y_1)*f^{-1}(y_2)$ for all $\displaystyle y_1$ and $\displaystyle y_2$ in B. I suggest you start by defining, for given $\displaystyle y_1$ and $\displaystyle y_2$ in B, $\displaystyle x_1= f^{-1}(y_1)$ and $\displaystyle x_2= f^{-1}(y_2)$.

3. ## Re: Inverse of a homomorphic function

I cant figure out how to use latex on this site? how is it done here?

4. ## Re: Inverse of a homomorphic function

Start with [ t e x ] and end with [ / t e x ]. A tutorial on using LaTeX is at
LaTeX Help

5. ## Re: Inverse of a homomorphic function

Originally Posted by mathrld
I cant figure out how to use latex on this site? how is it done here?
On the tool bar is a $\displaystyle \Sigma$ icon. That gives [TEX][/TEX] LaTeX wrap.
[TEX]\sqrt{x^4+1}[/TEX] gives $\displaystyle \sqrt{x^4+1}$.

Thanks DDD