There are two main theorems on the isomorphism of cyclic groups.

A. Any two cyclic groups of order n<∞ are isomorphic.

B. Any two infinite cyclic groups are isomorphic.

First I will partially prove A.

Any two cyclic groups of order n<∞ are isomorphic.

Proof:

Let G₁ and G₂ = , |G₁|= |G₂ |= n

Define f : G₁ →G₂ by f(a^k)=b^k

We will prove here that f is well-defined, 1-1 and that f “preserves the operation.”

Showing “well-defined”:

Suppose that a^k = a^t and that t>k. Thus: a^t-k = e . But then n|(t-k) . (Here I use the fact from chapter 2 that if o(a)=n and a^j = e then n|j.) But then b^t-k = e and hence b^k = b^t proving “well-defined. (Have we shown that k=t? NO!)

Showing 1-1 is similar:

If f(a^k )=f( a^t) then b^k = b^t and b^t-k = e. Whence a^t-k = e and a^k = a^t

Finally:

f(a^k a^t) = f(a^{k +t}) = b^{k +t}. OTOH f(a^k)=b^k and f(a^t)=b^t

Thus f(a^k)· f(a^t)= b^k · b^t = b^{k +t}

proving that f(a^k a^t) = f(a^k)· f(a^t)

To complete this proof all you need to do is show “onto”.

Now for theorem B I give a proof unlike the one in the text.

Let’s prove that: Every cyclic group is isomorphic to the integers under addition. Then B is a corollary of this proof.

Proof:

Let G an infinite cyclic group, G and Z be the integers. Define f:Z →G by f(k) = a^k . Showing that f is 1-1 and onto is an exercise.

And: f(k+j)= a^k+j = a^k a^j = f(k)·f(j) quickly shows that the operation is preserved.