The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8?(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Isomorphism! Help!

**Physics Forums | Science Articles, Homework Help, Discussion**