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  "body": "\n## Summary\n\nIn algebraic group theory, a **wonderful compactification** of a variety acted on by an algebraic group $G$ is a $G$-equivariant compactification such that the [[closure|0QEAJB91GZ]] of each orbit is smooth. (txt, 1–44) constructed a wonderful compactification of any symmetric variety given by a [[quotient|0KYX2N69J0]] $G/G^{\\sigma}$ of an algebraic group $G$ by the subgroup $G^{\\sigma}$ fixed by some involution $\\sigma$ of $G$ over the [[complex numbers|09FTAET5Z4]], sometimes called the **De Concini–Procesi compactification**. (txt) generalized this construction to arbitrary characteristic. In particular, by writing a group $G$ itself as a symmetric homogeneous space, $G=(G \\times G)/G$ (modulo the diagonal subgroup), this gives a wonderful compactification of the group $G$ itself.\n",
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