Answer to Question #300207 in Abstract Algebra for ksmoorthy

Question

If G is the abelian group of integers in the m apping T: G → G given by T(x ) = x then prove that as an autom orphism

Answer 1

Let us prove that the mapping $T: G → G$ given by $T(x ) = x$ is an automorphism.

Since $T(xy)=xy=T(x)T(y)$ for any $x,y\in G,$ we conclude that the mapping is a homomorphism. If $T(x)=T(y)$ then $x=y,$ and thus the mapping $T$ is injective. For any $y\in G,$ we have that $T(y)=y,$ and thus $T$ is a surjection. We conclude that $T$ is an automorphism.