P. Hauck’s research while affiliated with University of Tübingen and other places

... We take further previous research on the influence of two-generated subgroups on the structure of groups, in interaction with the study of products of subgroups. In [13] the following result is proven: (2) For all primes p = q, all p-elements a ∈ A and all q-elements b ∈ B, a, b is soluble. Obviously, for the special case A = B = G, the following well-known result of J. Thompson is derived: [18,6] ...

Reference:

Products of finite connected subgroups

... According to [8, Proposition 3.1], 2-generated connected involutory quandles are affine. Let Q be the involutory quandle SmallQuandle (15,6) of the RIG database. The quandle Q is latin and so its 2-generated subquandles are affine. ...

Reference:

On Core Quandles

... Structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [1,14,2]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [8,9]; in [10] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...

Reference:

Products of finite connected subgroups

... They have obtained the following nice result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble. In addition, they have also generalized the above mentioned result of Baer by replacing the condition of normality of A, B in G and using the following weaker condition: A permutes with all subgroups of B and B permutes with all subgroups of A. Their results in [3] were further developed and applied by many authors (see, for example, [1] [5][6][7][8], [14], [19]). We also notice that O.H.Kegel has also obtained many elegant results for soluble groups and supersoluble groups by considering the products of their subgroups (see [16][17][18]). ...

Reference:

Criterions of Supersolubility for Products of Supersoluble Groups

... There has been substantial research on characterizations of F-injector for various types of soluble Fitting classes F (see [5,6,7,9,10,12,14,15]). It is well known that the product of any two Fitting classes is also Fitting class and multiplication of Fitting classes satisfies associative law (see [3,Theorem IX.(1.12)(a),(c)]). ...

Reference:

On Injectors of a Hartley Set of a Finite Group

... The structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [7][8][9]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [10,11]; in [12] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...

Reference:

Products of Finite Connected Subgroups

... The structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [7][8][9]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [10,11]; in [12] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...

Reference:

Products of Finite Connected Subgroups

... The structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [7][8][9]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [10,11]; in [12] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...

Reference:

Products of Finite Connected Subgroups

... These factorized groups are said to be the product of the totally permutable subgroups A and B by R. Maier in[33]. Later on, a deep understanding of the structure of such groups has been reached and this study has been extended both in the frameworks of formation theory (see[7]–[10],[12],[15],[16],[33]) as well as in the theory of Fitting classes ([22]–[24]). A detailed account on this topic can be found in the book[5]. ...

Reference:

On conditional permutability and factorized groups

... But initially R. Maier in [24] proved that Asaad and Shaalan result is a particular case of a more general one when considering the class U of supersoluble groups as a saturated formation (containing U ). Later on it was proved that Maier's result extends to non-saturated formations which contain all supersoluble groups [6] and totally permutable products of groups have been deeply studied both in the frameworks of formation theory (see [7,10]) as well as in the theory of Fitting classes [17][18][19]. W. Guo, K.P. Shum and A.N. Skiba in [16] have extended previous results by considering a weaker condition of subgroups permutability, namely conditional permutability. More precisely they consider the following concepts: ...

Reference:

On finite products of groups and supersolubility