Nilpotent residual of a finite group

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Rodrigues E., de Melo E., ERCAN G.

Journal of Algebra, vol.641, pp.534-545, 2024 (SCI-Expanded, Scopus)

  • Publication Type: Article / Article
  • Volume: 641
  • Publication Date: 2024
  • Doi Number: 10.1016/j.jalgebra.2023.11.011
  • Journal Name: Journal of Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.534-545
  • Keywords: Automorphisms, Dihedral groups, Frobenius groups, Frobenius-like groups, Nilpotent residual
  • Open Archive Collection: AVESIS Open Access Collection
  • Middle East Technical University Affiliated: Yes

Abstract

Let F be a nilpotent group acted on by a group H via automorphisms and let the group G admit the semidirect product FH as a group of automorphisms so that CG(F)=1. We prove that the order of γ∞(G), the rank of γ∞(G) are bounded in terms of the orders of γ∞(CG(H)) and H, the rank of γ∞(CG(H)) and the order of H, respectively in cases where either FH is a Frobenius group; FH is a Frobenius-like group satisfying some certain conditions; or FH=〈α,β〉 is a dihedral group generated by the involutions α and β with F=〈αβ〉 and H=〈α〉.