How to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists? I have 2 groups $U_5$ and $U_{12}$ , .. 
$U_5 = \{1,2,3,4\}, U_{12} = \{1,5,7,11\}$. 
I have to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists.
I started with the "$yes$" case: there is an isomorphism.
So I searched an isomorphism $\varphi$ , but I didn't found. So I guess there is no an isomorphism $\varphi$  . 
How can I prove it? or at least explain? please help.

In $U_{12}$ everybpdy is his own inverse. This is not the case in $U_5$.
Along the same lines, $2$ is a generator of $U_5$, while $U_{12}$ has no generator. Since one group is cyclic and the other not, they cannot be isomorphic.
The same fact, restated: $U_{5}$ has an element of order $4$, while all elements of $U_{12}$ have order $1$ or $2$.
Remark: If you looked for an isomorphism, testing all viable possibilities, and nothing worked, a complete record of the testing is a proof of non-isomorphism.
However, a more cxommon way of showing non-isomorphism of two groups $A$ and $B$ is by using structural properties.
If you have a property $P$ that must be preserved by isomorphism, and $A$ has property $P$, but $B$ doesn't, then you know $A$ and $B$ cannot be isomorphic.
