A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of C_G(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.
ERCAN, GÜLİN; GÜLOĞLU, İSMAİL ŞUAYİP; and KHUKHRO, EVGENY
"Frobenius-like groups as groups of automorphisms,"
Turkish Journal of Mathematics: Vol. 38:
6, Article 2.
Available at: https://journals.tubitak.gov.tr/math/vol38/iss6/2