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This is a continuation of the expository article \cite{krp} with some newremarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbertspace $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and positioncovariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ bethe group of all unitary operators in $L^2 (\mathbb{R}^n)$ conjugations bywhich leave $S_n$ invariant. Here we prove the following results. $K_n$ is aclosed convex set for which a matrix $S$ is an extreme point if and only if$S=\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n,\mathbb{R}).$ Every element in $K_n$ is of the form $\frac{1}{2} (L^{T} L +M^{T} M)$ for some $L,M$ in $Sp (2n, \mathbb{R}).$ Every Gaussian state in $L^2(\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\mathbb{R}^{2n}).$Any element $U$ in the group $\mathcal{G}_n$ is of the form $U = \lambda W({\bm {\alpha}}) \Gamma (L)$ where $\lambda$ is a complex scalar of modulusunity, ${\bm {\alpha}} \in \mathbb{C}^n,$ $L \in Sp (2n, \mathbb{R}),$ $W({\bm{\alpha}})$ is the Weyl operator corresponding to ${\bm {\alpha}} $ and $\Gamma(L)$ is a unitary operator which implements the Bogolioubov automorphism of theLie algebra generated by the canonical momentum and position observablesinduced by the symplectic linear transformation $L.$

The Symmetry Group of Gaussian States in $L^2 (\mathbb{R}^n)$