0000000000121748
AUTHOR
Arnold D. Feldman
ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS
AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A sub…
Nilpotent length and system permutability
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.
Permutability of injectors with a central socle in a finite solvable group
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].
On generalised subnormal subgroups of finite groups
Let be a formation of finite groups. A subgroup M of a finite group G is said to be -normal in G if belongs to . A subgroup U of a finite group G is called a K--subnormal subgroup of G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ … ≤ Un = G such that Ui − 1 is either normal or -normal in Ui, for i = 1, 2, …, n. The K--subnormality could be regarded as the natural extension of the subnormality to formation theory and plays an important role in the structural study of finite groups. The main purpose of this paper is to analyse classes of finite groups whose K--subnormal subgroups are exactly the subnormal ones. Some interesting extensions of well-known classes of groups emerge.
A class of generalised finite T-groups
Let F be a formation (of finite groups) containing all nilpotent groups such that any normal subgroup of any T-group in F and any subgroup of any soluble T-group in F belongs to F. A subgroup M of a finite group G is said to be F-normal in G if G/CoreG(M) belongs to F. Named after Kegel, a subgroup U of a finite group G is called a K- F-subnormal subgroup of G if either U=G or U=U0?U1???Un=G such that Ui?1 is either normal in Ui or Ui1 is F-normal in Ui, for i=1,2,...,n. We call a finite group G a TF-group if every K- F-subnormal subgroup of G is normal in G. When F is the class of all finite nilpotent groups, the TF-groups are precisely the T-groups. The aim of this paper is to analyse the…
Injectors with a central socle in a finite solvable group
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).
Injectors with a normal complement in a finite solvable group
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.