  In 1980 J. Powell proposed that five specific elements sufficed to generate
the Goeritz group of any Heegaard splitting of $S^3$, extending work of Goeritz
on genus $2$ splittings. Here we prove that Powell's conjecture was correct for
splittings of genus $3$ as well, and discuss a framework for deciding the truth
of the conjecture for higher genus splittings.
