On Generalized Wielandt Subgroup (original) (raw)

In this paper, we define a generalized Wielandt subgroup, local generalized Wielandt subgroup and its series for finite group and discuss its different basic properties which explain the notion of generalized Wielandt subgroup in a better way. We bound generalized Wielandt length as a function of nilpotency classes of its Sylow subgroups.

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Generalisations of the Wielandt subgroup

2016

Die Wielandt-Untergruppe und die Norm einer Gruppe sind gruppentheoretische Konzepte, die als Verallgemeinerungen des Zentrums einer Gruppe eingefuhrt wurden. Das erste Ziel dieser Arbeit ist es die Ideen von Wielandt weiter zu verallgemeinern. Zu diesem Zweck untersuchen wir die strukturellen Eigenschaften der bereits in meiner MPhil-Arbeit eingefuhrten Verallgemeinerung. Zu einer gegebenen Gruppe G wird diese Verallgemeinerung relativ zu einem Normalteiler in G definiert. Wir untersuchen die strukturellen Eigenschaften dieser verallgemeinerten Wielandt-Untergruppe und die Lange der verallgemeinerten Wielandt-Reihe. Auserdem verbessern wir einen Fehler in der MPhil-Arbeit. Wir fuhren eine neue Verallgemeinerung der Wielandt-Untergruppe ein, die wir "relative Wielandt-Untergruppe" nennen. Wir bemerken, dass diese relative Wielandt-Untergruppe einige Eigenschaften hat, die fur die gewohnliche Wielandt-Untergruppe nicht gelten. Zusatzlich geben wir ein Beispiel, welches zeig...

A generalisation of a theorem of Wielandt

Journal of Algebra

In 1974, Helmut Wielandt proved that in a finite group G, a subgroup A is subnormal if and only if it is subnormal in every A, g for all g ∈ G. In this paper, we prove that the subnormality of an odd order nilpotent subgroup A of G is already guaranteed by a seemingly weaker condition: A is subnormal in G if for every conjugacy class C of G there exists c ∈ C for which A is subnormal in A, c. We also prove the following property of finite non-abelian simple groups: if A is a subgroup of odd prime order p in a finite almost simple group G, then there exists a cyclic p ′-subgroup of F * (G) which does not normalise any non-trivial p-subgroup of G that is generated by conjugates of A.

A Note on Finite Metabelian Groups of Wielandt Length Two

Southeast Asian Bulletin of Mathematics, 2001

The class of groups of Wielandt length one has been extensively studied and recently the class of groups of Wielandt length two has been studied, in particular nilpotent groups of odd order by E.A. Ormerod and supersoluble groups of order prime to six by A. Ali. In this paper we consider metabelian groups of odd order and provide characterizations for those groups with nilpotent residual either a Hall subgroup or of prime power order.

On a Class of Generalized Nilpotent Groups

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

The nilpotent length of finite soluble groups

Journal of the Australian Mathematical Society, 1973

References Chapter 1. Preliininaries 1.1. General results. The Frattini subgroup of a group G, denoted by <I'(G), is the intersection of all the maximal subgroups of G. An element ^G is said to be omissible in G if, whenever <g,X> = G for some subset X of G, then G =<x). We now state som.e well-known results concerning the Frattini subgroup, v;hich will be used frequently in the sequel. The proofs of these results can all be found in Section 3.3 of [7]. Lemma 1.1.1 (a) An element x of G lies in ^ (.G) if and only if x is omissible in G. (b) If N G then G has a proper subgroup H such that NH = G if and only if N ^ 9(G). (c) The Frattini subgroup of a group is nilpotent. (d) If G has a normal subgroup N ^ (G) such that G/N is nilpotent, then G is itself nilpotent.

Wielandt’s results for algebraic k-groups

Journal of Lie Theory, 2006

We analyze the relation between subnormality and nilpotence, the subnormal joint property, some criteria of subnormality, the norm and the Wielandt subgroup in the case of algebraic groups defined over an arbitrary field.

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