Answer to Question #289460 in Abstract Algebra for ramya

Question

If G is the abelian group of integers in the mapping T: G → G given by T(x ) = x then prove that as an automorphism

Answer 1

Solution:

Let $G$ be the abelian group of integers. Let us show that the mapping $T: G → G$ given by $T(x ) = x$ is an automorphism.

Since $T(x+y)=x+y=T(x)+T(y),$ we conclude that $T$ is a homomorphism.

Since for $x\ne y$ we get that $T(x)=x\ne y=T(y),$ we conclude that $T$ is one-to-one.

Taking into account that for any $y\in G$ we have that $T(y)=y,$ we conclude that $T$ is surjective.

Therefore, $T$ is a bijection, and hence is an automorphism.