Why do the symmetries of the monster vertex algebra form a finite simple group?

Together with their 1988 construction of the monster vertex algebra $V^\natural$, Frenkel, Lepowsky, and Meurman showed that the largest sporadic simple group, known as the Fischer-Griess monster, forms the symmetry group of an infinite dimensional algebraic object whose construction was motivated by theoretical physics. However, the fact that the symmetry group is in fact finite and simple ultimately relied on highly non-trivial group-theoretic results used in Griess's work on the monster. We prove some properties of the automorphism group of $V^\natural$, most notably that it is is finite and simple, using recent developments in the theory of vertex operator algebras, but mostly 19th century group theory.