Left and right quotient sets in non-abelian groups

Abstract: For a group $G$, we define the right quotient set and the left quotient set as follows: $$ AA^{-1}:=\{a_1a_2^{-1}:a_1,a_2\in A\} \qquad A^{-1}A:=\{a_1^{-1}a_2:a_1,a_2\in A\}.$$ We examine the relationships between the left and right quotient sets. If $G$ is an abelian group, then these sets are equal, but subtleties arise in non-abelian settings, as these sets may not have the same cardinality. Tao remarked that the cardinality difference $|AA^{-1}| - |A^{-1}A|$ may be arbitrarily large for certain groups.

We first give explicit constructions of sets $A$ where this difference attains every possible integer, proving that the difference can be any possible value if $G$ has elements of order 2.

We also find the minimum cardinality of $A$ 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 a graph called the difference graph $D_A$ that encodes equality in the right quotient set. Similarly, $D_{A^{-1}}$ encodes equality in the left quotient set. By observing an isomorphism of edges in $D_A$ and $D_{A^{-1}}$ and counting connected components, we are able to prove the results above. In the free group on two generators, we can prove that the difference $|AA^{-1}| - |A^{-1}A|$ is always even. We explicitly construct subsets of $F_2$ that achieve every even integer. In the infinite dihedral group $D_\infty \cong \mathbb{Z} \rtimes \mathbb{Z}/2$, we prove that every integer difference is achievable, using the results of Martin and O'Bryant on the cardinality differences of sum sets and difference sets in $\mathbb{Z}.$

Joint work with June Duvivier, Xiaoyao Huang, Ava Kennon, Say-yeon Kwon, Steven J. Miller, Arman Rysmakhanov, and Ren Watson

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)