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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 
$RC{A_0}$ (or even over 
$RCA_0^{\rm{*}}$) 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.
