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