The author gives a proof of the theorem that for Coxeter groups the algebra of coinvariants is isomorphic to the normalizer of the Coxeter group in the linear group of a vector space over \(\mathbb{R}\). First he proves the existence of a homomorphism of groups from the normalizer of \(G\) in \(GL(V)\), \(N_{GL(V)}(G)\), to \(\text{Aut}(P(V)_ G)\), called \(\Phi^*\), which is injective if \(G\) is a Coxeter group. Next, since \(P(V)_ 1 \cong (P(V)_ G)_ 1 \cong V\), each \(P(V)_ G\)-automorphism \(A\) induces an element of the linear group of \(V\). Each of these is in \(N_{GL(V)}(G)\). Thus \(\Phi^*\) is surjective. Finally, the main result is achieved: if \(G\) is a Coxeter group, then \(N_{GL(V)}(G) \cong \text{Aut}(P(V)_ G)\). The proof given is rather elementary, requiring only elementary algebra, some group theory, and a little knowledge of invariants.