Answer to Question #214482 in Abstract Algebra for Priya

Question

For abelian group, identity mapping is:
(a)one-one (b)onto
(C)homomorphism (d)all of the above

Answer 1

Let us consider Abelian group $G$ and its identity mapping $\varphi: G\to G,\ \varphi(x)=x.$ If $x_1\ne x_2,$ then $\varphi(x_1)=x_1\ne x_2=\varphi(x_2),$ and hence the mapping is one-to-one. For any $y\in G$ we have that $\varphi(y)=y,$ and consequently, $\varphi$ is onto. Since $\varphi(xy)=xy=\varphi(x)\varphi(y)$ for any $x,y\in G,$ we conclude that the mapping is a homomorphism.

Answer: d