- #1
k3N70n
- 67
- 0
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
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