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

Step-by-step explanation:

G is the abelian group.we want to prove that it is an automorpism.

Answer 2

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

Given :

G is the abelian group of integers in the mapping T : G → G given by T(x) = x

To find :

To prove T is an automorphism.

Solution :

Step 1 of 2 :

Write down the given mapping

Here it is given that G is the abelian group of integers in the mapping T : G → G given by

T(x) = x

Step 2 of 2 :

Prove that T is an automorphism.

Injective :

Let x , y ∈ G such that

T(x) = T(y)

⇒ x = y

So T is injective

Surjective :

Let x ∈ G

Now T(x) = x

So T is Surjective

Thus T is bijective

Homomorphism :

Let x , y ∈ G and c ∈ F

Now T(cx + y)

= cx + y [ By definition of given mapping ]

= cT(x) + T(y)

So T is Homomorphism

Hence T is an automorphism.

Hence the proof follows

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