On One Open Question of the Theory of

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-Properties of a Finite Group

Abstract

Let

\sigma =\{{\sigma }_i\mid i\in I\}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma =\{\sigma _{i} \mid i\in I\}$$\end{document}

be some partition of the set of all primes and G a finite group. A subgroup A of G is

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-permutable in G provided G is

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-full; that is, G has a Hall

{\sigma }_i

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{i}$$\end{document}

-subgroup for all

i\in I

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i\in I$$\end{document}

and A permutes with all such Hall subgroups H of G; that is,

AH=HA

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AH=HA$$\end{document}

. Answering the Question 6.4 in Skiba (Probl Phys Math Tech 42(21):89–96, 2014), we get a description of finite

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-full groups G in which

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-permutability is a transitive relation.

Keywords

Finite group /

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-soluble group / Weak Robinson

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-complex /

\sigma

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}

-permutable subgroup /

P\sigma T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\sigma T$$\end{document}

-group / 20D10 / 20D15 / 20D30

References

[1] Adarchenko NM. A new characterization of finite σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-soluble PσT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\sigma T$$\end{document}-groups. Algebra Discrete Math., 2020, 29(1): 33-41

[2] Agrawal RK. Finite groups whose subnormal subgroups permute with all Sylow subgroups. Proc. Amer. Math. Soc., 1975, 47: 77-83

[3] Ballester-Bolinches A, Ezquerro LMClasses of Finite groups, 2006DordrechtSpringer

[4] Ballester-Bolinches A, Pedraza-Aguilera MC, Pèrez-Calabuing V. On two classes of generalized T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T$$\end{document}-groups. Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales. Serie A. Matematicas, 2023, 117: 105

[5] Ballester-Bolinches A, Beidleman JC, Heineken H. Groups in which Sylow subgroups and subnormal subgroups permute, Special issue in honor of Reinhold Baer (1902–1979). Illinois J. Math., 2003, 47(1–2): 63-69

[6] Ballester-Bolinches A, Esteban-Romero R, Asaad MProducts of Finite Groups, 2010Berlin-New YorkWalter de Gruyter

[7] Doerk K, Hawkes TFinite Soluble Groups, 1992Berlin-New YorkWalter de Gruyter

[8] Guo WStructure Theory for Canonical Classes of Finite Groups, 2015Heidelberg-New York-Dordrecht-LondonSpringer

[9] Guo W, Skiba AN. On σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-supersoluble groups and one generalization of CLT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$CLT$$\end{document}-groups. J. Algebra, 2018, 512: 92-108

[10] Gorenstein D Finite Simple Groups. An Introduction to Their Classification, 1982New YorkPlenum Press

[11] Huppert BEndliche Gruppen I, 1967Berlin-Heidelberg-New YorkSpringer-Verlag

[12] Huppert B, Blackburn NFinite Groups III, 1982Berlin, New-YorkSpringer-Verlag

[13] Hu B, Huang J, Skiba AN. On two open problems of the theory of permutable subgroups of finite groups. Publ. Math. Debrecen, 2019, 94(3–4): 477-491

[14] Knyagina BN, Monakhov VS. On π′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi ^{\prime }$$\end{document}-properties of finite groups having a Hall π\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi $$\end{document}-subgroup. Siberian Math. J., 2011, 52(2): 234-243

[15] Li H , Liu A-M, Safonova IN, Skiba AN. Characterizations of some classes of finite σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-soluble PσT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\sigma T$$\end{document}-groups. Commun. Algebra, 2023

[16] Liu A-M, Guo W, Safonova IN, Skiba AN. G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G$$\end{document}-covering subgroup systems for some classes of σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-soluble groups. J. Algebra, 2021, 585: 280-293

[17] Robinson DJS. The structure of finite groups in which permutability is a transitive relation. J. Austral. Math. Soc., 2001, 70: 143-159

[18] Skiba AN. On σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-properties of finite groups I. Probl. Phys. Math. Techn., 2014, 4(21): 89-96

[19] Skiba AN. On σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-subnormal and σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-permutable subgroups of finite groups. J. Algebra, 2015, 436: 1-16

[20] Skiba AN. On some results in the theory of finite partially soluble groups. Commun. Math. Stat., 2016, 4(3): 281-309

[21] Skiba AN. Some characterizations of finite σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-soluble PσT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\sigma T$$\end{document}-groups. J. Algebra, 2018, 495(1): 114-129

[22] Skiba AN. On sublattices of the subgroup lattice defined by formation Fitting sets. J. Algebra, 2020, 550: 69-85

[23] Weinstein M Between Nilpotent and Solvable, 1982EdinburghPolygonal Publishing House

[24] Zhang X-F, Guo W, Safonova IN, Skiba AN. A Robinson description of finite PσT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P\sigma T$$\end{document}-groups. J. Algebra, 2023, 631: 218-235

[25] Zhu X , Cao C, Guo W. Finite σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-soluble groups in which σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-permutability is a transitive relation. J. Algebra Appl., 2019, 18(4): 1950064

Funding

National Natural Science Foundation of China(12171126)

Belarusian Republican Foundation for Fundamental Research(F23RNF-237)

Ministry of Education of the Republic of Belarus(20211328)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature