Noncoprime action of a cyclic group

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ERCAN G., Güloğlu İ. Ş.

Journal of Algebra, cilt.643, ss.1-10, 2024 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 643
  • Basım Tarihi: 2024
  • Doi Numarası: 10.1016/j.jalgebra.2023.12.020
  • Dergi Adı: Journal of Algebra
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.1-10
  • Anahtar Kelimeler: Automorphism, Fixed point free action, Nilpotent length
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ(A), the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2ℓ(A) if |G| is odd. More generally we prove that the nilpotent length of G is at most 2ℓ(A)+c(G;A) when G is of odd order and A normalizes a Sylow system of G where c(G;A) denotes the number of trivial A-modules appearing in an A-composition series of G.