Frobenius action on Carter subgroups

2019/07/25 by Gülin Ercan, İsmail Ş. Güloğlu, Ercan, Gülin +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1907.10951

arxiv created 2019/07/25 · arxiv updated 2019/07/26

Abstract

Let G be 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. 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 π, the terms of the π-series of C_G(H) is obtained as the intersection of C_G(H) with the corresponding terms of the π-series of G, and the π-length of G may exceed the π-length of C_G(H) by at most one. They generalize the main results of \citeKhu.

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