Show that $U(8)$ is Isomorphic to $U(12)$.

Question

Question: Show that $U(8)$ is Isomorphic to $U(12)$

The groups are:

$U\left ( 8 \right )=\left \{ 1,3,5,7 \right \}$

$U\left ( 12 \right )=\left \{ 1,5,7,11 \right \}$

I think there is a bit of subtle point that I am not fully understanding about isomorphism which is hindering my progress. The solution mentions about the order of an element but I do not understand how that is pivotal to solving this.

Thanks in advance.

Answer 1

Let $F:U(8) \rightarrow U(12)$ be such that $F(1)=1´; F(3)=11´; F(5)=5´; F(7)=7´$. Note that $ F$ is bijective, to show that $F$ is an isomorphism we only need to show that $F$ is indeed operation preserving.Observe

$F(1*n)=F(n) = F(n)*1'=F(n)*F(1)$ for all $n \in U(8)$; $F(3*5)=F(7)=7'=11'*5'=F(3)*F(5);\\ F(5*7)=F(35)=F(3)=11´=5´*7´=F(5)*F(7); \\ F(3*7)=F(21)=F(5)=5´=11´*7´=F(3)*F(7). $

(Where $*$ reffers for respective group operations)

$\Rightarrow F$ is a homomorphism & F is bijective $\Rightarrow F$ is an isomorphism $\Rightarrow U (8) \cong U(12).$