Answer to Question #285570 in Abstract Algebra for Aysh

Question #285570

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

1
Expert's answer
2022-01-10T11:37:47-0500

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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS