What does it mean for two groups to be locally isomorphic? I am reading Zee's QFT book and he repeatedly remarks that one group is locally isomorphic to the direct product or sum of some other groups.  As I read it, it "local isomorphism" basically means "equal."  What symbol is used to denote local isomorphism latex?  Is there some other kind of group isomorphism besides local?  Wikipedia uses the symbol $\cong$ for isomorphism and it says that for two groups $(G,*)$ and $(H,\odot)$, the groups are isomorphic if there exists a function such that for any $u,v\in G$, we have
$$ f(u*v)=f(u)\odot f(v)  .$$
Is this Zee's "local isomorphism?"  For the groups $SO(3)$ and $SU(2)$, we have $(\mathbb{O}_3,\times)$ and $(\mathbb{U}_2,\times)$ where $\mathbb{O}_3$ are $3\times3$ real orthogonal matrices with unit determinant and $\mathbb{U}_2$ are $2\times2$ unitary matrices with the same determinant.  What $f$ allows me to demonstrate the isomorphism?
 A: For any pair of Lie groups $G$ and $H$, they will necessarily have associated Lie algebras, $\mathfrak{g}$ and $\mathfrak{h}$. Note that it is a general property from Lie group theory that the Lie algebra of some Lie group is isomorphic to the tangent space at the identity of the Lie group, as viewed as a manifold.
Now, in general, it may be the case that the Lie algebras are isomorphic, meaning $\mathfrak{g}\simeq \mathfrak{h}$. This does not imply that the Lie groups themselves are isomorphic to each other.
But since the tangent spaces of the two Lie groups are isomorphic, it makes sense to say that the groups are locally isomorphic, meaning there exists an isomorphism (which is also a diffeomorphism) between coordinate patches of the two Lie groups, but there is no such global map.
The most common example of this is $SU(2)$ and $SO(3)$. These are locally isomorphic in the sense described -- they both have the same Lie algebra, namely the algebra associated with spin. Globally, however, $SU(2)$ can be shown to be the double-cover of $SO(3)$. This is, in fact, the reason why $SO(3)$ has only integer spin representations despite its Lie algebra supporting half-integer representations.
