Subindices of two-element subsets of finite groups

Abstract

Let $A$ be a nonempty subset of a group $G$. A right subfactor relative to $A$ is a subset $B$ maximal under inclusion among those for which every element of $AB$ has a unique representation as $ab$, with $a\in A$ and $b\in B$. Left subfactors are defined analogously using $BA$. The infima and suprema of the cardinalities of right and left subfactors define four \emph{subindices}. A subset is called \emph{index-stable} if its four subindices coincide. For any infinite group $G$, every nonempty finite subset of $G$ is index-stable. For finite groups, the situation is more complicated. We prove that for a two-element subset of a finite group, the left and right subindices coincide pairwise. Let $A=\{x,y\}$ with $x\ne y$, and let $X$ be the cyclic subgroup generated by $x^{-1}y$. Then the lower subindex of $A$ is $|G:X|\cdot\lceil |X|/3\rceil$, and the upper subindex is $|G:X|\cdot\lfloor |X|/2\rfloor$. These formulas imply that $A$ is index-stable if and only if $|X|$ is one of $2,3,4,5,7$. For any two-element subset of a finite group $G$, the lower subindex is at least $\lceil |G|/3\rceil$, and the upper subindex is at most $\lfloor |G|/2\rfloor$. We establish necessary and sufficient conditions for equality in each of these bounds. All two-element subsets of a finite group are index-stable if and only if every element of the group has order $1$, $2$, $3$, $4$, $5$, or $7$. We give a complete structural classification of such groups.