0000000000358693

AUTHOR

Rex Dark

showing 4 related works from this author

Nilpotent length and system permutability

2022

Abstract If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

CombinatoricsMathematics::Group TheoryMaximal subgroupNilpotentFinite groupClass (set theory)Algebra and Number TheoryConjugacy classGroup (mathematics)Sylow theoremsBasis (universal algebra)MathematicsJournal of Algebra
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Permutability of injectors with a central socle in a finite solvable group

2017

In response to an Open Question of Doerk and Hawkes [5, IX Section 3, page 615], we shall show that if Zπ is the Fitting class formed by the finite solvable groups whose π-socle is central (where π is a set of prime numbers), then the Zπ-injectors of a finite solvable group G permute with the members of a Sylow basis in G. The proof depends on the properties of certain extraspecial groups [4].

Class (set theory)Algebra and Number Theory010102 general mathematicsSylow theoremsPrime numberBasis (universal algebra)01 natural sciencesFitting subgroupSet (abstract data type)CombinatoricsSection (category theory)Solvable group0103 physical sciences010307 mathematical physics0101 mathematicsMathematicsJournal of Algebra
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Injectors with a central socle in a finite solvable group

2013

Abstract In response to an Open Question of Doerk and Hawkes (1992) [2, IX §4, p. 628] , we shall describe three constructions for the Z π -injectors of a finite solvable group, where Z π is the Fitting class formed by the finite solvable groups whose π -socle is central (and π is a set of prime numbers).

Class (set theory)Algebra and Number Theoryfitting classinjectorPrime numberFitting subgroupCombinatoricsSet (abstract data type)Soclecentral socleSolvable groupfinite solvable group theoryNilpotent groupMathematics
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Injectors with a normal complement in a finite solvable group

2011

Abstract Suppose G is a finite solvable group, and H is a subgroup with a normal complement in G. We shall find necessary and sufficient conditions (some of which are related to the properties of coprime actions) for H to be an injector in G. We shall also use these criteria to find characterizations of injectors which need not have a normal complement.

AlgebraAlgebra and Number TheoryCoprime integersSolvable groupinjectorfitting setfinite solvable group theorynormal complementComplement (complexity)Mathematics
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