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Orders of Automorphism Groups of Finite Groups

  • Inder Bir Singh PassiEmail author
  • Mahender Singh
  • Manoj Kumar Yadav
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

The object of study in this chapter is the relation between the order of a finite group and that of its group of automorphisms. In 1954, Scott [114] conjectured that a finite group has at least a prescribed number of automorphisms if the order of the group is sufficiently large. The conjecture was confirmed by Ledermann and Neumann [80, Theorem 6.6] in 1956 by constructing an explicit function \(f: \mathbb {N} \rightarrow \mathbb {N}\) with the property that if the finite group G has order \(|G |\ge f(n)\), then \(|{\text {Aut}} (G)| \ge n\). In the same year, building on the techniques from [80], the authors [81] proved the following local version of Scott’s conjecture: Conjecture 3.1. There exists a function \(f:\mathbb {N} \rightarrow \mathbb {N}\) such that for each \(h \in \mathbb {N}\) and each prime p, if G is any finite group such that \(p^{f(h)}\) divides |G|, then \(p^h\) divides \(|{\text {Aut}} (G)|\). Later on, Green [49], Howarth [63] and Hyde [68] successively improved the function f to a quadratic polynomial function. The aim of this chapter is to give an exposition of these developments. Schur multiplier plays a significant role in these investigations.

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Inder Bir Singh Passi
    • 1
    Email author
  • Mahender Singh
    • 2
  • Manoj Kumar Yadav
    • 3
  1. 1.Centre for Advanced Study in MathematicsPanjab UniversityChandigarhIndia
  2. 2.Department of Mathematical SciencesIndian Institute of Science Education and Research MohaliSAS NagarIndia
  3. 3.School of MathematicsHarish-Chandra Research Institute HBNIJhunsi AllahabadIndia

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