On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”
Earlier, the author described up to conjugacy all pairs 𝐴 𝐵 of nilpotent subgroups of a finite group 𝐺 with socle subscript 𝐿 2 𝑞 for which 𝐴 superscript 𝐵 𝑔 1 for any element of 𝐺 . A similar description was obtained by the author later for primary subgroups 𝐴 and 𝐵 of a finite group 𝐺 with so...
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| Published in | Proceedings of the Steklov Institute of Mathematics Vol. 323; no. Suppl 1; pp. S321 - S332 |
|---|---|
| Main Author | |
| Format | Journal Article Conference Proceeding |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.12.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0081-5438 1531-8605 |
| DOI | 10.1134/S0081543823060251 |
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| Abstract | Earlier, the author described up to conjugacy all pairs
𝐴
𝐵
of nilpotent subgroups of a finite group
𝐺
with socle
subscript
𝐿
2
𝑞
for which
𝐴
superscript
𝐵
𝑔
1
for any element of
𝐺
. A similar description was obtained by the author later for primary subgroups
𝐴
and
𝐵
of a finite group
𝐺
with socle
subscript
𝐿
𝑛
superscript
2
𝑚
. In this paper, we describe up to conjugacy all pairs
𝐴
𝐵
of nilpotent subgroups of a finite group
𝐺
with simple socle from the “Atlas of Finite Groups” for which
𝐴
superscript
𝐵
𝑔
1
for any element
𝑔
of
𝐺
. The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group
𝐺
for any nilpotent subgroups
𝑁
contains an element
𝑔
such that
𝑁
superscript
𝑁
𝑔
1
. |
|---|---|
| AbstractList | Earlier, the author described up to conjugacy all pairs
𝐴
𝐵
of nilpotent subgroups of a finite group
𝐺
with socle
subscript
𝐿
2
𝑞
for which
𝐴
superscript
𝐵
𝑔
1
for any element of
𝐺
. A similar description was obtained by the author later for primary subgroups
𝐴
and
𝐵
of a finite group
𝐺
with socle
subscript
𝐿
𝑛
superscript
2
𝑚
. In this paper, we describe up to conjugacy all pairs
𝐴
𝐵
of nilpotent subgroups of a finite group
𝐺
with simple socle from the “Atlas of Finite Groups” for which
𝐴
superscript
𝐵
𝑔
1
for any element
𝑔
of
𝐺
. The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group
𝐺
for any nilpotent subgroups
𝑁
contains an element
𝑔
such that
𝑁
superscript
𝑁
𝑔
1
. Earlier, the author described up to conjugacy all pairs ð´ðµ of nilpotent subgroups of a finite group ðº with socle subscriptð¿2ð' for which ð´superscriptðµð'"1 for any element of ðº. A similar description was obtained by the author later for primary subgroups ð´ and ðµ of a finite group ðº with socle subscriptð¿ð'>superscript2ð'. In this paper, we describe up to conjugacy all pairs ð´ðµ of nilpotent subgroups of a finite group ðº with simple socle from the “Atlas of Finite Groups” for which ð´superscriptðµð'"1 for any element ð'" of ðº. The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group ðº for any nilpotent subgroups ð' contains an element ð'" such that ð'superscriptð'ð'"1. |
| Author | Zenkov, V. I. |
| Author_xml | – sequence: 1 givenname: V. I. surname: Zenkov fullname: Zenkov, V. I. email: v1i9z52@mail.ru organization: Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal University |
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| Cites_doi | 10.17377/semi.2017.14.121 10.1112/plms/s2-1.1.388 10.17377/semi.2018.15.058 10.1134/S0001434619030076 10.1007/978-1-4684-8497-7 10.1134/S0037446618020088 10.1007/BF02547953 10.1090/S0002-9939-1963-0142631-X 10.1017/S0027763000017438 10.1007/BF02110750 10.1017/CBO9781139192576 10.1134/S0037446616060070 10.1134/S0037446613010102 10.1017/S0013091500004065 10.1007/s10958-017-3232-8 10.1007/s10469-020-09603-x 10.1134/S0081543814050228 10.1112/plms/s2-2.1.432 10.1007/BF02757885 10.1007/BF01158735 10.17516/1997-1397-2018-11-2-171-177 10.1017/S0013091500026328 10.1016/0021-8693(71)90086-X 10.1007/BF02367025 10.1007/BF01877481 10.1007/BF02789326 10.21538/0134-4889-2021-27-1-70-78 10.17377/semi.2016.13.087 10.1134/S0081543816090182 10.1007/BF02287057 10.1016/j.jalgebra.2010.09.024 |
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| Keywords | nilpotent subgroup finite group intersection of subgroups Fitting subgroup |
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| References | Burnside (CR7) 1904; 2 Alperin, Lyons (CR35) 1971; 19 Mazurov, Khukhro (CR15) 2010 Gorenstein (CR16) 1982 Zenkov (CR30) 2016; 13 Zenkov, Nuzhin (CR26) 2018; 11 Zenkov (CR6) 2016; 295 Bialostocki (CR14) 1975; 20 Kabanov, Makhnev, Starostin (CR12) 1976; 15 Laffey (CR4) 1977; 20 Harada, Yamaki (CR38) 1979; 1 Aschbacher, Seitz (CR31) 1976; 63 Zenkov (CR11) 1996; 2 Dempwolff, Wong (CR5) 1982; 25 Brodkey (CR3) 1963; 14 Zenkov (CR24) 2018; 15 Burnside (CR8) 1905; 2 Kurmazov (CR17) 2013; 54 Bray, Holt, Roney-Dougal (CR36) 2013 Herzog (CR13) 1972; 11 Zenkov (CR20) 2021; 27 Isaacs (CR2) 2008 Zenkov (CR28) 2018; 59 Zenkov (CR21) 2019; 105 Zenkov (CR1) 1994; 56 Zenkov (CR19) 2020; 59 Baer (CR33) 1957; 133 Mazurov, Zenkov (CR10) 1996; 35 Zenkov, Nuzhin (CR27) 2017; 221 Yang (CR32) 2011; 325 Conway, Curtis, Norton, Parker, Wilson (CR23) 1985 Gorenstein, Lyons, Solomon (CR29) 1998 Zenkov (CR25) 2017; 14 Kondrat’ev (CR37) 1985; 37 Ito (CR9) 1958; 9 Zenkov (CR18) 2014; 285 Zenkov (CR22) 2016; 57 Suzuki (CR34) 1968; 82 D Gorenstein (8389_CR16) 1982 VI Zenkov (8389_CR18) 2014; 285 N Bialostocki (8389_CR14) 1975; 20 VI Zenkov (8389_CR24) 2018; 15 (8389_CR15) 2010 VI Zenkov (8389_CR26) 2018; 11 VI Zenkov (8389_CR19) 2020; 59 VD Mazurov (8389_CR10) 1996; 35 VI Zenkov (8389_CR11) 1996; 2 JS Brodkey (8389_CR3) 1963; 14 VI Zenkov (8389_CR27) 2017; 221 JN Bray (8389_CR36) 2013 N Ito (8389_CR9) 1958; 9 VI Zenkov (8389_CR1) 1994; 56 VI Zenkov (8389_CR20) 2021; 27 VV Kabanov (8389_CR12) 1976; 15 R Baer (8389_CR33) 1957; 133 M Suzuki (8389_CR34) 1968; 82 W Burnside (8389_CR7) 1904; 2 VI Zenkov (8389_CR22) 2016; 57 M Aschbacher (8389_CR31) 1976; 63 J Alperin (8389_CR35) 1971; 19 JH Conway (8389_CR23) 1985 VI Zenkov (8389_CR21) 2019; 105 VI Zenkov (8389_CR6) 2016; 295 RK Kurmazov (8389_CR17) 2013; 54 U Dempwolff (8389_CR5) 1982; 25 M Herzog (8389_CR13) 1972; 11 Yong Yang (8389_CR32) 2011; 325 D Gorenstein (8389_CR29) 1998 VI Zenkov (8389_CR30) 2016; 13 AS Kondrat’ev (8389_CR37) 1985; 37 IM Isaacs (8389_CR2) 2008 TJ Laffey (8389_CR4) 1977; 20 W Burnside (8389_CR8) 1905; 2 VI Zenkov (8389_CR25) 2017; 14 VI Zenkov (8389_CR28) 2018; 59 K Harada (8389_CR38) 1979; 1 |
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| Snippet | Earlier, the author described up to conjugacy all pairs
𝐴
𝐵
of nilpotent subgroups of a finite group
𝐺
with socle
subscript
𝐿
2
𝑞
for which
𝐴... Earlier, the author described up to conjugacy all pairs ð´ðµ of nilpotent subgroups of a finite group ðº with socle subscriptð¿2ð' for which... |
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| SubjectTerms | Group theory Mathematics Mathematics and Statistics Subgroups |
| Title | On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups” |
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