Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / a / Solution

A Lie group is a smooth manifold $G$ equipped with smooth maps $m:G\times G\to G$, $m(g,h)=gh$, and $\iota :G\to G$, $\iota (g)=g^{-1}$, for which multiplication is associative, there is an identity $e$, and every $g$ has the inverse $g^{-1}$. In equations,Thus the group axioms and the smooth structure are compatible.