On the Determination of Sylow Numbers
Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Kimmerle, Wolfgang [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Indian National Science Academy 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Indian journal of pure and applied mathematics - New Delhi : Acad., 1970, 52(2021), 3 vom: Sept., Seite 652-668 |
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| Übergeordnetes Werk: |
volume:52 ; year:2021 ; number:3 ; month:09 ; pages:652-668 |
| Links: |
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| DOI / URN: |
10.1007/s13226-021-00183-9 |
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| Katalog-ID: |
SPR050380192 |
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| 520 | |a Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. | ||
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| 650 | 4 | |a Integral group rings |7 (dpeaa)DE-He213 | |
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10.1007/s13226-021-00183-9 doi (DE-627)SPR050380192 (SPR)s13226-021-00183-9-e DE-627 ger DE-627 rakwb eng Kimmerle, Wolfgang verfasserin aut On the Determination of Sylow Numbers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 Köster, Iris aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 52(2021), 3 vom: Sept., Seite 652-668 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:52 year:2021 number:3 month:09 pages:652-668 https://dx.doi.org/10.1007/s13226-021-00183-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 52 2021 3 09 652-668 |
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10.1007/s13226-021-00183-9 doi (DE-627)SPR050380192 (SPR)s13226-021-00183-9-e DE-627 ger DE-627 rakwb eng Kimmerle, Wolfgang verfasserin aut On the Determination of Sylow Numbers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 Köster, Iris aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 52(2021), 3 vom: Sept., Seite 652-668 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:52 year:2021 number:3 month:09 pages:652-668 https://dx.doi.org/10.1007/s13226-021-00183-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 52 2021 3 09 652-668 |
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10.1007/s13226-021-00183-9 doi (DE-627)SPR050380192 (SPR)s13226-021-00183-9-e DE-627 ger DE-627 rakwb eng Kimmerle, Wolfgang verfasserin aut On the Determination of Sylow Numbers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 Köster, Iris aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 52(2021), 3 vom: Sept., Seite 652-668 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:52 year:2021 number:3 month:09 pages:652-668 https://dx.doi.org/10.1007/s13226-021-00183-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 52 2021 3 09 652-668 |
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10.1007/s13226-021-00183-9 doi (DE-627)SPR050380192 (SPR)s13226-021-00183-9-e DE-627 ger DE-627 rakwb eng Kimmerle, Wolfgang verfasserin aut On the Determination of Sylow Numbers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 Köster, Iris aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 52(2021), 3 vom: Sept., Seite 652-668 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:52 year:2021 number:3 month:09 pages:652-668 https://dx.doi.org/10.1007/s13226-021-00183-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 52 2021 3 09 652-668 |
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10.1007/s13226-021-00183-9 doi (DE-627)SPR050380192 (SPR)s13226-021-00183-9-e DE-627 ger DE-627 rakwb eng Kimmerle, Wolfgang verfasserin aut On the Determination of Sylow Numbers 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 Köster, Iris aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 52(2021), 3 vom: Sept., Seite 652-668 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:52 year:2021 number:3 month:09 pages:652-668 https://dx.doi.org/10.1007/s13226-021-00183-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 52 2021 3 09 652-668 |
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Enthalten in Indian journal of pure and applied mathematics 52(2021), 3 vom: Sept., Seite 652-668 volume:52 year:2021 number:3 month:09 pages:652-668 |
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In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. 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Kimmerle, Wolfgang |
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Kimmerle, Wolfgang misc Character tables misc Integral group rings misc Sylow numbers On the Determination of Sylow Numbers |
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On the Determination of Sylow Numbers Character tables (dpeaa)DE-He213 Integral group rings (dpeaa)DE-He213 Sylow numbers (dpeaa)DE-He213 |
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On the Determination of Sylow Numbers |
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10.1007/s13226-021-00183-9 |
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on the determination of sylow numbers |
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On the Determination of Sylow Numbers |
| abstract |
Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. © The Indian National Science Academy 2021 |
| abstractGer |
Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. © The Indian National Science Academy 2021 |
| abstract_unstemmed |
Abstract Let G be a finite group and denote by %${\mathbb {Z}}G%$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided %${\mathbb {Z}}G \cong {\mathbb {Z}}H%$ and G is q-constrained. If additionally %$O_{q'}(G)%$ is soluble the set %$\mathrm{sn}(G)%$ of all Sylow numbers of G is determined by %${\mathbb {Z}}G.%$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table %$\mathrm{X}(G)%$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components %${\mathbb {Z}}G%$ determines %$\mathrm{sn}(G).%$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated. © The Indian National Science Academy 2021 |
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On the Determination of Sylow Numbers |
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|
| score |
7.401105 |