Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

Abstract

We prove that a finite solvable group G admitting a Frobenius group F H of automorphisms of coprime order with kernel F and complement H so that [G, F ] = G and C C G (F ) (h) = 1 for every 1 = h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

“…Previously, it was proved in [9] that if C G B = 1, then many properties of G are "close" to the corresponding properties of C G A , possibly also depending on A. It is natural to try to extend these results to more general situations, for example, the condition C G B = 1 was relaxed in [3,10] under certain additional conditions, and the condition of BA being Frobenius was replaced in [4] by BA/ B B being Frobenius with B B of prime order, but keeping the condition C G B = 1.…”

“…In the second product on the right, each element z x = c y 1 x c y n x belongs to C Q W 1 , since x ∈ A (since A ∩ X = E ∩ X) and c y i x ∈ C Q W 1 for all i by (3). For each element in the first product on the right of (8), we have z y ≡ z mod C Q W 1 for any y ∈ E. Indeed, the image of z in Q/C Q W 1 is also central and therefore is represented by a scalar linear transformation in the action on the homogeneous Wedderburn component W 1 for Q: also note that W 1 and therefore C Q W 1 are Einvariant, so that the action of y ∈ X ∩ E on the image of z in Q/C Q W 1 is well defined.…”

Action of a Frobenius-like group with fixed-point free kernel

“…Notice that the action of H on the set of F -orbits on is transitive, and K Ä C Q .H /. Hence K is trivial on the whole of V contrary to (2). Thus H 1 ¤ 1.…”

“…In contrast to [2] one cannot even weaken the condition C VQ .F / D 1 to the condition C C VQ .F / .h/ D 1 for all nonidentity elements h 2 H .…”

“…There has been a lot of research on the structure of solvable groups admitting a Frobenius group FH of automorphisms (see [2,[5][6][7]). In particular under some additional hypothesis it was shown that various properties of G, for instance the nilpotent length of G, are close to those of C G .H /.…”