Why is $U(10)\not\approx U(12)$? I'm trying to understand this example on isomorphism but failing to do so. I know that if two groups have a differing number of elements of each order they are not isomorphic. I assume this is what they are saying. Particularly, how is it that $\phi(3)\phi(3)$ and $\phi(1)\phi(1)$ both equal 1?


As written explicitly, in $U(12)$ we have $g^2=1$ for every element $g$. Hence, $\phi(3)^2=\phi(1)^2=1$.
However, what you wrote about differing numbers of elements of the same order is indeed a simpler way to think of it. In $U(10)$ there are elements of order $4$, and in $U(12)$ there are no such elements. Consequently, the two groups cannot be isomorphic to one another.
