This is in relation to Kodaira's Complex Manifolds and Deformation Complex Structures Chpt 2, Sec 2.
$W$ is a complex Lie group. A discrete subgroup $G\leq W$ gives properly discontinuous and fixed point free action on $W$. Thus $W/G$ makes sense as a complex manifold. Now the book says $W/G$ is a complex Lie group as well without mentioning $G$ normal.
$\textbf{Q:}$ Why does $W/G$ inherits a group structure? Note that I need $W\times W\xrightarrow{\cdot} W$ descends to the quotient level map. From standard group theory, $W/G$ is group iff $G$ is normal by considering $wG\cdot 1G=wG$. The natural procedure is to assume $G$ is normal which forces $G\leq Z(W)$ if $W$ is connected where $Z(W)$ is the center. However, the book did not mention $G$ being normal. Where does normality coming from then or have I missed something here?
I can't comment on the book, since I don't have access to it now. But I can tell you that the statement is false. Just take, for instance$$\left\{\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}1&0\\0&-1\end{bmatrix}\right\},$$which is a non-normal subgroup of $GL(2,\mathbb C)$.