Group Theory

Question

Prove that :-

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

Comments

nicely explained here https://math.stackexchange.com/questions/1752570/infinite-cyclic-group-is-isomorphic-to-the-group-of-integers-under-addition

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Thank you !

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Do we have these in our Current GATE syllabus?

Answer 1

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) = a^n+m = aⁿa^m = 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.

Comments

it is a representation like power set

not multiplication

rt?

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Here the operator on G need not to be multiplication

It can be anything say # defined in such a way that G forms an infinite cyclic group

In group theory a^m means a is operated on itself m times. i.e. a # a # a ....(m times)

Now, f(n+m) = a^(n+m) (i.e. a#a#a....(m+n times)) = a^m # a^n = f(m) # f(n).

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got it... thank you !