Proving U(8) is not Isomorphic to U(10): Insights and Techniques

[k3N70n]

Hi. Hoping a could have a little bit of guidance with this question

Show that U(8) is not isomorphic to U(10)

So far, I've realized that in U(8) each element is it's own inverse while in U(10) 3 and 7 are inverses of each other. I guess that's really all I need to say that they aren't isomorphic but my suspicion is that I should be stating this in a more formal way then a simple Cayley table.

Previously, I worked out a somewhat similar question where I had to find the automorphisms of Z_4. I said:
let f:Z_4 --> Z_4
and f(0)=0 (because the identity must be mapped to the identity by a theorem early proved)
then f(2) = f(1) + f(1)
f(3) = f(1) + f(2)

So then we have 4 cases for f(1)...[went on to show that if f(1) = 0 or 2 then f was not injective]

I was thinking something similar here may be appropriate but I'm not sure how to set it up. Thanks in advance for any help

 

[HallsofIvy]

I see nothing wrong with saying exactly what you did. If there exist an isomorphism from U(10) to U(8), then f(7) and f(3) must map into inverses: but what ever f(10) is, its inverse is itself: f(7) must equal f(3) contradicting the fact that it is an isomorphism.

 

[k3N70n]

Thanks HallsofIvy. I guess I was just trying to make the question harder than it really was.