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SUMMARY:Pramana Saldin (University of California\, Berkeley)
DTSTART:20260716T150000Z
DTEND:20260716T152500Z
DTSTAMP:20260710T111633Z
UID:CANT2026/47
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/47/
 ">Left and right quotient sets in non-abelian groups</a>\nby Pramana Saldi
 n (University of California\, Berkeley) as part of Combinatorial and addit
 ive number theory seminar (CANT 2026)\n\nLecture held in Science Center in
  the CUNY Graduate Center (4th floor).\n\nAbstract\nFor a group $G$\, we d
 efine the right quotient set and the left quotient set as follows: $$\n AA
 ^{-1}:=\\{a_1a_2^{-1}:a_1\,a_2\\in A\\} \\qquad A^{-1}A:=\\{a_1^{-1}a_2:a_
 1\,a_2\\in A\\}.$$ \nWe examine the relationships between the left and rig
 ht 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. \n\nWe firs
 t give explicit constructions of sets $A$ where this difference attains ev
 ery possible integer\, proving that the difference can be any possible val
 ue if $G$ has elements of order 2. \n\nWe also find the minimum cardinalit
 y of $A$ so that the difference between the cardinalities of the left and 
 right quotient sets is nonzero\, depending on the existence of order $2$ e
 lements in $G$. \n\nTo prove these results\, we construct a graph called t
 he difference graph $D_A$ that encodes equality in the right quotient set.
  Similarly\, $D_{A^{-1}}$ encodes equality in the left quotient set. By ob
 serving an isomorphism of edges in $D_A$ and $D_{A^{-1}}$ and counting con
 nected components\, we are able to prove the results above. In the free gr
 oup 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 achie
 ve 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 cardinali
 ty differences of sum sets and difference sets in $\\mathbb{Z}.$ \n\nJoint
  work with June Duvivier\, Xiaoyao Huang\, Ava Kennon\, Say-yeon Kwon\, St
 even J. Miller\, Arman Rysmakhanov\, and Ren Watson\n
LOCATION:https://researchseminars.org/talk/CANT2026/47/
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