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3 results

  • 2 - GroupTheory

  • Esko Keski-Vakkuri, University of Helsinki, Claus Montonen, University of Helsinki, Marco Panero, Università degli Studi di Torino, Italy
  • Book:
    Mathematical Methods for Physics
    Published online:
    08 December 2022
    Print publication:
    22 December 2022, pp 4-51

  • Summary

    After defining the basic concepts of group theory, we discuss some particularly interesting finite groups. Then, we introduce the formalism of Young diagrams and an alternative definition of groups through their presentation. The final part of the chapter is devoted to continuous groups and to groups acting on a set.

  • REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

  • KOSTAS HATZIKIRIAKOU, STEPHEN G. SIMPSON
  • Journal:
    The Journal of Symbolic Logic / Volume 82 / Issue 2 / June 2017
    Published online by Cambridge University Press:
    19 June 2017, pp. 576-589
    Print publication:
    June 2017

  • Let S be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of S over a field K of characteristic 0. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over $RC{A_0}$ (or even over $RCA_0^{\rm{*}}$) to the statement that ${\omega ^\omega }$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

  • A Constructive Proof of Brauer's Theorem on Induced Characters in the Group Ring

  • Fedor Bogomolov, Frederick P. Greenleaf
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 57 / Issue 1 / February 2014
    Published online by Cambridge University Press:
    18 December 2013, pp. 31-53
    • Article
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  • We provide an alternative constructive proof of the classical Brauer theorem for finite groups based on the well-known description of the complex irreducible representations of the symmetric groups Sn. The theorem is first proved for Sn and then for general G by embedding in Sn and applying the Mackey subgroup theorem.