Answer to Question #295628 in Abstract Algebra for chinchu

Question #295628

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


1
Expert's answer
2022-02-10T03:47:10-0500

Solution:

Let "G" be the abelian group of integers. Let us show that the mapping "T: G \u2192 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.


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