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→G_{1} 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 a^{n} =e iff n=0.
Define f:Z→G by f(n) = a^{n} for all n∈Z.
Clearly f is a surjective function and f(n+m) = a^{n+m} = a^{n}a^{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 | a^{n} = 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.