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Prove that :-

Every infinite cyclic group is isomorphic to the infinite cyclic group of integers under addition.

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I hope that you know the definitions of Homomorphism and Monomorphism of groups

To, solve the above problem you have to know one theorem

Theorem  : A homomorphism f : G→G1 of groups is a monomorphism iff Ker f = {e}

Now, come to the problem

Let, G = <a> be an infinite cyclic group. Then o(a) is not finite and this implies an =e iff n=0.

Define f:Z→G by f(n) = an for all n∈Z.

Clearly f is a surjective function and f(n+m) = an+m = anam = f(n)f(m) (Definition of Homomorphism of groups). for all n,m∈Z

Now, Ker f = { n∈Z | f(n) = e } = { n∈Z | an = e } = {0}

This implies f is injective (Theorem written above).

Thus f is an isomorphism.

So, the infinite cyclic group is isomorphic to the set of integers under addition.

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