Another question about $SU(2) / \Bbb Z_2$

Question

As is well-known, a lot of scripts use the isomorphism or "identity":

$$SU(2)/\Bbb Z_2 = SO(3)$$

I understand that a lot of people write the equal sign but mean isomorphic to... However, what I read elsewhere is:

$$SU(2) / \{-1,1\} = SO(3)$$

However, the quotient ring $ \Bbb Z_2 = \Bbb Z/2 \Bbb Z = \{0,1\} $ and not $\{-1,1\}$. Here it is not clear to me, if the first equation is a generally accepted error, another isomorphism to the 2nd equation I don't understand or something else I am missing. Please don't down-vote, I am struggling with this a lot...

Edit: An even correcter version to me seems to be

$$ SU(2) / \{ -I, I \} = SO(3) $$, where $I$ is the identity matrix in $SU(2)$.

Answer 1

The additive group $\mathbb{Z}_2=\{0,1\}$ is isomorphic to the multiplicative group $\mathbb{Z}^{\times}=\{-1,1\}$.

Answer 2

People like to write $X/Y$, even when $Y$ isn't an actual subgroup or action or equivalence relation of $X$, to mean $X/Z$, where $Z$ is that one "obvious" or "unambiguous" subgroup/action/equivalence of $X$ that looks just like $Y$.

Examples:

• $SU(2)/\mathbb Z_2$ actually means $SU(2)/\{I, -I\}$, as you've noticed already.

• $SO(3)/SO(2)$ denotes the $2$-sphere, even though $SO(2)$ is not a subgroup of $SO(3)$. However, the subgroup of matrices $\begin{pmatrix}*&*&0\\*&*&0\\0&0&1\end{pmatrix}$ of $SO(3)$ looks just like $SO(2)$, so that's what's actually meant by that notation.