A cycle-tiling criterion characterizes when a multiset A in a finite nonabelian group G can be realized as quotients from two bijections, with the abelianization product condition shown insufficient even when the product in G is the identity, via a counterexample in S3.
Hall Jr., A combinatorial problem on abelian groups,Proc
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A nonabelian twist on differences of bijections
A cycle-tiling criterion characterizes when a multiset A in a finite nonabelian group G can be realized as quotients from two bijections, with the abelianization product condition shown insufficient even when the product in G is the identity, via a counterexample in S3.