4. Let $n>0, n=2^{k} m$ (where $m$ is odd). Then the greatest common divisor of $\mathrm{C}_{2 n}^{1}, \mathrm{C}_{2 n}^{3}$, $\cdots, \mathrm{C}_{2 n}^{2 n-1}$ is $\qquad$
4. $2^{k+1}$.

First, $2^{k+1} \| \mathrm{C}_{2 n}^{1}$.
It is easy to prove that $2^{k+1} \| \mathrm{C}_{2 n}^{i} (i=3,5, \cdots, 2 n-1)$. This is because $i \mathrm{C}_{2 n}^{i}=2 n \mathrm{C}_{2 n-1}^{i-1}, (i, 2 n)=1$.
Therefore, the greatest common divisor of $\mathrm{C}_{2 n}^{1}, \mathrm{C}_{2 n}^{3}, \cdots, \mathrm{C}_{2 n}^{2 n-1}$ is $2^{k+1}$.