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

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

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

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

Therefore, TT is a bijection, and hence is an automorphism.


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