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Groups with Divisibility Property-I

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

Abstract

Every finite non-cyclic abelian p-group of order greater than \(p^2\) has the property that its order divides that of its group of automorphisms (Theorem  3.34). The problem whether every non-abelian p-group of order greater than \(p^2\) possesses the same property has been a subject of intensive investigation. As discussed in the introduction, this property is referred to as the Divisibility Property. While several classes of p-groups have been shown to have Divisibility Property, it is now known that not all finite p-groups admit this property [46]. An exposition of these developments is presented in the remaining part of this monograph. In this chapter, some reduction results, due to Buckley [14], are presented in Sect. 4.1. Among other results, it is proved that one can confine attention to the class of purely non-abelian p-groups. In subsequent sections it is shown that Divisibility Property is satisfied by p-groups of nilpotency class 2 [33], p-groups with metacyclic central quotient [18], modular p-group [22], p-abelian p-groups [19], and groups with small central quotient [20]. In view of Theorem  3.34, it can be assumed that the groups under consideration are non-abelian p-groups. The main ingredient in verifying Divisibility Property for various classes of groups G is the subgroup
$${\text {IC}} (G):={\text {Inn}} (G){\text {Autcent}} (G)$$
of the automorphism group \({\text {Aut}} (G)\) of G.

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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