1. group homomorphism

If $\displaystyle a:G\to H$ is a group homomorphism between G and H, then we have the following definition:

for all $\displaystyle g,k\in G$ we have $\displaystyle a(g*k)=a(g)*a(k)$

According to wikipedia, this means that $\displaystyle a(1_G)=1_H$, where $\displaystyle 1_G,1_H$, are the identity elements of G and H respectively, and also that $\displaystyle a(g^{-1})=a(g)^{-1}$ for all $\displaystyle g\in G$.

Can someone show me how these conditions are true?

Thanks

2. Hi

One way is to write $\displaystyle a(1_G)=a(1_G1_G)=a(1_G)a(1_G)$ thus $\displaystyle a(1_G)=1_H.$

Now that we know that, $\displaystyle a(g^{-1})a(g)=a(g^{-1}g)=a(1_G)=1_H$ implies that $\displaystyle a(g^{-1})=(a(g))^{-1}.$

3. Thanks, but I don't understand how your first argument shows that $\displaystyle a(1_G)=1_H$

4. Ok, that's because a group law is strong enough: let $\displaystyle A$ be a group, $\displaystyle e$ its identity element and $\displaystyle a\in A,$ then $\displaystyle a^2=a\Rightarrow a=e$
Indeed, $\displaystyle a^2=a\Rightarrow a=a^{-1}a^2=a^{-1}a=e$

That may be a particular case of: $\displaystyle \forall a,b\in A,\ \exists !x\ \text{s.t.}\ ax=b$

Indeed if $\displaystyle b=a,$ since $\displaystyle e$ is a solution, by unicity, $\displaystyle aa=a\Rightarrow a=e$

5. Thanks, I got it now.