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If $G$ is a complex simply connected semisimple algebraic group and if$\lambda$ is a dominant weight, we consider the compactification $X_\lambda$ inthe projectivisation of $\End(V(\lambda))$ obtained as the closure of the$G\times G$-orbit of the identity and we give necessary and sufficientconditions on the support of $\lambda$ so that $X_\lambda$ is normal; as well,we give necessary and sufficient conditions on the support of $\lambda$ so that$X_\lambda$ is smooth.

Normality and non-normality of group compactifications in simple projective spaces