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Updated: Feb 22, 2026

Murine Drinking Models in the Development of Pharmacotherapies for Alcoholism: Drinking in the Dark and Two-bottle Choice
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Pariah moonshine.

John F R Duncan1, Michael H Mertens2, Ken Ono3

  • 1Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA, 30322, USA. john.duncan@emory.edu.

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|September 23, 2017
PubMed
Summary
This summary is machine-generated.

Finite simple groups are key to understanding symmetry. This study reveals hidden symmetries in pariah groups, connecting them to number theory and deep mathematical problems.

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Area of Science:

  • Group Theory
  • Number Theory
  • Algebraic Geometry

Background:

  • Finite simple groups are fundamental to understanding symmetry.
  • The classification of finite simple groups led to the discovery of 'pariah' groups, which are not part of natural families.
  • Monstrous moonshine revealed unexpected connections between the 'monster' group, number theory, and physics.

Purpose of the Study:

  • To investigate if a phenomenon similar to monstrous moonshine exists for pariah groups.
  • To uncover hidden symmetries associated with pariah groups in number theory and related fields.
  • To explore the role of pariah groups in addressing profound mathematical challenges.

Main Methods:

  • Focusing on the O'Nan pariah group.
  • Analyzing its connections to quadratic forms and elliptic curves.
  • Proving congruences for class numbers, Selmer groups, and Tate-Shafarevich groups of elliptic curves.

Main Results:

  • The O'Nan pariah group exhibits hidden symmetries within quadratic forms and elliptic curves.
  • New congruences for class numbers of elliptic curves have been established.
  • The study demonstrates a link between pariah groups and deep problems in number theory and algebraic geometry.

Conclusions:

  • Pariah groups, previously considered isolated, possess significant hidden symmetries.
  • This research establishes a form of 'pariah moonshine,' linking these groups to number theory and elliptic curves.
  • The findings underscore the importance of pariah groups in contemporary mathematics and suggest potential links to physics.