Comparing Left and Right Quotient Sets in Groups

Abstract

For a finite subset $A$ of a group $G$, we define the right quotient set and the left quotient set of $A$, respectively, as $AA^{-1} := \{a_1a_2^{-1}:a_1,a_2\in A\}$, $A^{-1}A := \{a_1^{-1}a_2:a_1,a_2\in A\}$. While the right and left quotient sets are equal if $G$ is abelian, subtleties arise when $G$ is a nonabelian group, where the cardinality difference $|AA^{-1}| - |A^{-1}A|$ may be take on arbitrarily large values. Using the results of Martin and O'Bryant on the cardinality differences of sum sets and difference sets in $\mathbb{Z}$, we prove in the infinite dihedral group, $D_\infty \cong \mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$, every integer difference is achievable. Further, we prove that in $F_2$, the free group on $2$ generators, an integer difference is achievable if and only if that integer is even, and we explicitly construct subsets of $F_2$ that achieve every even integer. We further determine the minimum cardinality of $A \subset G$ so that the difference between the cardinalities of the left and right quotient sets is nonzero, depending on the existence of order $2$ elements in $G$. To prove these results, we construct difference graphs $D_A$ and $D_{A^{-1}}$ which encode equality, respectively, in the right and left quotient sets. We observe a bijection from edges in $D_A$ to edges in $D_{A^{-1}}$ and count connected components in order to obtain our results on cardinality differences $|AA^{-1}| - |A^{-1}A|$.