I have seen that if $G$ is a finite group and $H$ is a proper subgroup of $G$ with finite index then $ G \neq \bigcup\limits\_{g \in G} gHg^{-1}$. Does this remain true for the infinite case also?



Group cannot be the union of conjugates
Not in general. Every matrix in $\text{GL}\_2(\mathbf C)$ is conjugate to an invertible upper triangular matrix (use eigenvectors), and the invertible upper triangular matrices are a proper subgroup.


