  Let $S$ be the group of finitely supported permutations of a countably
infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of
characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$
satisfies the ascending chain condition for two-sided ideals. We study the
reverse mathematics of this theorem, proving its equivalence over RCA$_0$ (or
even over RCA$_0^*$) to the statement that $\omega^\omega$ is well ordered. Our
equivalence proof proceeds via the statement that the Young diagrams form a
well partial ordering.
