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Derived length of a Frobenius-like kernel.
- Source :
-
Journal of Algebra . Aug2014, Vol. 412, p179-188. 10p. - Publication Year :
- 2014
-
Abstract
- Abstract: A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that is a Frobenius group with Frobenius kernel . Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide . It is proved that the derived length of the kernel F is bounded solely in terms of the dimension of the fixed-point subspace of H by . It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most , where r is the sectional rank of . As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to , a bound for the p-length of G is obtained in terms of the rank of a Hall -subgroup of . Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic). [Copyright &y& Elsevier]
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 412
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 96407106
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2014.04.025