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,
$$
(gh)k=g(hk),\qquad eg=ge=g,\qquad gg^{-1}=g^{-1}g=e.
$$
Thus the <group axioms> and the smooth structure are compatible.

Solution

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

{wiki=Group_(mathematics)#Definition}

The group axioms require closure and associativity of the binary operation, an identity element, and an inverse for every element.


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 Incoming links: Group axioms