# Proof that homomorphism of an inverse equals the inverse of the homomorphism

• Oct 14th 2011, 10:57 AM
MSUMathStdnt
Proof that homomorphism of an inverse equals the inverse of the homomorphism
\$\displaystyle g\$ is an element of group \$\displaystyle G\$. \$\displaystyle f(g)\$ is a homomorphism from \$\displaystyle G\$ to \$\displaystyle H\$. Prove that \$\displaystyle f(g^{-1})=f(g)^{-1}\$.

I'm told the proof is by multiplying the two terms, like this (I guess)
\$\displaystyle f(g^{-1}) f(g)^{-1} = ??\$
but still cannot see what to do.

I also have that \$\displaystyle e_G, e_H\$, are the identities in \$\displaystyle G\$ and \$\displaystyle H\$, respectively.
• Oct 14th 2011, 12:12 PM
HallsofIvy
Re: Proof that homomorphism of an inverse equals the inverse of the homomorphism
Oops, sorry. It is not "inverse of a homomorphism" but "homomorphism of an inverse".

That is, if ab= e1, the identity in the first group (ring, field?), then you want to show that f(a)f(b)= e2. Well, the whole point of a homomorphism is that f(ab)= f(a)f(b). What does that tell you if ab= e1?
• Oct 14th 2011, 12:53 PM
MSUMathStdnt
Re: Proof that homomorphism of an inverse equals the inverse of the homomorphism
Quote:

Originally Posted by HallsofIvy
That is, if ab= e1, the identity in the first group (ring, field?), then you want to show that f(a)f(b)= e2. Well, the whole point of a homomorphism is that f(ab)= f(a)f(b). What does that tell you if ab= e1?

\$\displaystyle g\$ in \$\displaystyle G\$ implies \$\displaystyle g^{-1}\$ in \$\displaystyle G\$.
\$\displaystyle f(g)*f(g^{-1})=f(g*g^{-1})=f(e1)=e2\$

Then just left multiply the first and last element by \$\displaystyle f(g)^{-1}\$.
• Apr 19th 2013, 02:47 PM
mohammedlabeeb
Re: Proof that homomorphism of an inverse equals the inverse of the homomorphism
How can I post things by using this beautiful notation?
• Apr 19th 2013, 03:05 PM
emakarov
Re: Proof that homomorphism of an inverse equals the inverse of the homomorphism
Quote:

Originally Posted by mohammedlabeeb
How can I post things by using this beautiful notation?

See this post.