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Cosubnormality and the Hypercenter
... Knapp proves in [5] the following characterisation of strong cosubnormality in terms of the hypercentre (Z ∞ (G) denotes the hypercentre of a group G). Theorem 1 [5,Theorem 3.3]. ...
... Knapp proves in [5] the following characterisation of strong cosubnormality in terms of the hypercentre (Z ∞ (G) denotes the hypercentre of a group G). Theorem 1 [5,Theorem 3.3]. Let A, B be subgroups of a group G. Then the following are equivalent: ...
Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join 〈A,B〉 and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in 〈A,B〉 and, if Z is the hypercentre of G=〈A,B〉, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal.
... Remark. The concept of N -connectedness is related to the concept of strong cosubnormality, introduced by Knapp [14] and defined as follows: ...
Two subgroups H and K of a finite group G are said to be N-connected if the subgroup generated by x and y is a nilpotent group, for every pair of elements x in H and y in K. This paper is devoted to the study of pairwise N-connected and permutable products of finitely many groups, in the framework of formation and Fitting class theory.
... Remark. The concept of N -connectedness is related to the concept of strong cosubnormality, introduced by Knapp [14] and defined as follows: Definition ([14, Definition 3.1]). Let G be a group and let A, B be subgroups of G. ...
Two subgroups H and K of a finite group G are said to
be -connected if the subgroup generated by x and
y is a nilpotent group, for every pair of elements x in
H and y in K. This paper is devoted to the study of
pairwise -connected and permutable products of
finitely many groups, in the framework of formation and
Fitting class theory.
... Remark. The concept of N -connectedness is related to the concept of strong cosubnormality, introduced by Knapp [14] and defined as follows: ...
For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.
Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.
A subgroup of a group G is called quasinormal in G if it permutes with every subgroup of G. For a finite group G it is well known that quasinormal subgroups are not far from being normal: It6 and Sz6p [3] have shown that a quasinormal subgroup of G containing no nontrivial normal subgroup of G is always nilpotent. In this paper we shall prove a stronger result, namely: Theorem. If Q is a quasinormal subgroup of the finite group G, then Q~/QG is contained in the hypercentre Zoo (G/Q~) of G/Q~.
Subnormale Untergruppen endlicher Gruppen, Vorlesung an der Universität Tübingen im Sommersemester 1971
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