Number of Isomorphisms f from G to G' of Order 8 Cyclic Groups

[sutupidmath]

The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8?

My thoughts:

Since f is an isomorphism, we know that it prserves the identity, so f:e-->e', e identity in G, e' identity in G'.

Also f preserves the order of each element. That is if o(a)=k=>o(f(a))=k

SO, i thought that f will send the el of the same order in G to the corresponding elements of the same order in G'.

Let G=[a], and G'=. so it means that there are 4 el in G that have order 8 ( the generators of G, a, a^3, a^5,a^7), so there are 4 possibilities for these elements, hense by keeping the other el. fixed we would have 4^4 isomorphisms.

But also we have 2 el of order 4, (a^2, and a^6) so there are two possibilities for these elements to be mapped into G' by f, so if the other el are fixed we would have 2^2 mappings.

Does this mean that the total nr of such isomorphisms is 4^4+2^2? Or am i totally on the wrong way?

Any suggestions would be appreciated.

 

[gel]

f must satisfy f(a)=b^k, some 1 <= k <= 7, and f is completely determined by the choice of k.
You should be able to see how many possible choices there are for k.