A nonabelian twist on differences of bijections

Abstract

Hall's theorem on differences of bijections characterizes the multisets $$ \{a_1,\ldots,a_{|G|}\} $$ in a finite abelian group $G$ that can be written in the form $$ a_i=b_i-c_i, $$ where both $b_1,\ldots,b_{|G|}$ and $c_1,\ldots,c_{|G|}$ are enumerations of $G$. The necessary and sufficient condition is the zero-sum condition $$ a_1+\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, $$ where $b$ and $c$ are bijections onto $G$. Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of $A$ is trivial in $ G_{\rm ab}. $ We show that this condition is not sufficient in general, even when the elements of $A$ admit an ordering whose product is the identity in $G$. The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of $A$ into product-one words whose partial-product sets tile $G$ by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in $ S_3, $ and we extend the same obstruction to infinitely many finite nonabelian groups.