Is there a group $G$ with the property that $G$ is a smooth manifold, the multiplication map of $G$ is smooth, but the inversion map of $G$ is not smooth?



Group G hasn't all conditions of Lie group
[Robert L. Bryant "An Introduction to Lie Groups and Symplectic Geometry"](https://www.math.duke.edu/~bryant/ParkCityLectures.pdf) requires in the definition of a Lie group only that the multiplication map be smooth, and then proves that the inversion map must be smooth also. (Proposition 1, page 14.)


