Frobenius action on Carter subgroups


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Ercan G. , Güloğlu İ. Ş.

INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, cilt.30, ss.1073-1080, 2020 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 30
  • Basım Tarihi: 2020
  • Doi Numarası: 10.1142/s0218196720500319
  • Dergi Adı: INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
  • Sayfa Sayıları: ss.1073-1080

Özet

Let G he a finite solvable group and H be a subgroup of Aut(G). Suppose that there exists an H-invariant Carter subgroup F of G such that the semidirect product FH is a Frobenius group with kernel F and complement H. We prove that the terms of the Fitting series of C-G (H) are obtained as the intersection of C-G (H) with the corresponding terms of the Fitting series of G, and the Fitting height of G may exceed the Fitting height of C-G (H) by at most one. As a corollary it is shown that for any set of primes pi, the terms of the pi-series of C-G (H) are obtained as the intersection of C-G (H) with the corresponding terms of the pi-series of G, and the pi-length of C may exceed the pi-length of C-G (H) by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1-11] obtained by Khukhro.