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On Supersolubility of a Group with Seminormal Subgroups

Authors:
V. S. Monakhov at Francisk Skorina Gomel State University
V. S. Monakhov
A. A. Trofimuk at Brest State University named after A.S. Pushkin
A. A. Trofimuk
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Abstract

A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that GU=(G′)NG^{\mathfrak{U}}=\left(G^{\prime}\right)^{\mathfrak{N}}. Moreover, if the indices of the subgroups A and B of G are coprime then GU=(GN)2G^{\mathfrak{U}}=\left(G^{\mathfrak{N}}\right)^{2}. Here N\mathfrak{N}, U\mathfrak{U}, and N2\mathfrak{N}^2 are the formations of all nilpotent, supersoluble, and metanilpotent groups respectively, while HXH^{\mathfrak{X}} is the X\mathfrak{X}-residual of H. We also prove the supersolubility of G = AB when all Sylow subgroups of A and B are seminormal in G.

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... The groups G = AB with seminormal supersoluble subgroups A and B were investigated in [5]. It is established that G U = (G ) N [5,Theorem 2.3]. ...
... The groups G = AB with seminormal supersoluble subgroups A and B were investigated in [5]. It is established that G U = (G ) N [5,Theorem 2.3]. The supersolubility of G = AB is also proved provided that all Sylow subgroups of A and B are seminormal in G [5,Theorem 2.4]. ...
... It is established that G U = (G ) N [5,Theorem 2.3]. The supersolubility of G = AB is also proved provided that all Sylow subgroups of A and B are seminormal in G [5,Theorem 2.4]. ...
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Let G be a group. A subgroup H is weakly subnormal in G if H=⟨A,B⟩ H=\langle A,B\rangle for some subnormal subgroup A and seminormal subgroup B in G . Note that B is seminormal in G if there exists a subgroup Y such that G=BY and AX is a subgroup for every subgroup X in Y . We give some new properties of weakly subnormal subgroups and new information about the structure of the group G=AB with weakly subnormal subgroups A and B . In particular, we prove that if A,B∈F A,B\in\mathfrak{F} , then GF≤(G′)N G^{\mathfrak{F}}\leq(G^{\prime})^{\mathfrak{N}} , where F \mathfrak{F} is a saturated formation such that U⊆F \mathfrak{U}\subseteq\mathfrak{F} . Here N \mathfrak{N} and U \mathfrak{U} are the formations of all nilpotent and supersoluble groups correspondingly, and GF G^{\mathfrak{F}} is the F {\mathfrak{F}} -residual of G . Moreover, we study the groups G=AB whose Sylow (maximal) subgroups from A and B are weakly subnormal in G .
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... The factorizable groups G = AB with w-supersoluble factors A and B were investigated in [8], [10], [11], [14]. There are many other papers devoted to study factorizable groups, and the reader is referred to the book [1] and the bibliography therein. ...
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Preprint
  • Feb 2020
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  • PUBL MATH-DEBRECEN
Let H and T be subgroups of a group G. Then we call H conditionally permutable (or in brevity, c-permutable) with T in G if there exists an element x ∈ G such that HT x = T x H. If H is c-permutable with T in < H, T >, then we call H completely c-permutable with T in G. By using the above concepts, we will give some new criterions for the supersolubility of a finite group G = AB, where A and B are both supersoluble groups. In particular, we prove that a finite group G is supersoluble if and only if G = AB, where both A, B are nilpotent subgroups of the group G and B is completely c-permutable in G with every term in some chief series of A. We will also give some applications of our new criterions.
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