On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups
Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor o...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Cao, Chenchen [verfasserIn] Guo, Wenbin [verfasserIn] Zhang, Chi [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Monatshefte für Mathematik - Wien [u.a.] : Springer, 1890, 189(2018), 2 vom: 07. Juni, Seite 239-242 |
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| Übergeordnetes Werk: |
volume:189 ; year:2018 ; number:2 ; day:07 ; month:06 ; pages:239-242 |
| Links: |
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| DOI / URN: |
10.1007/s00605-018-1201-z |
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| Katalog-ID: |
SPR007216033 |
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| 245 | 1 | 0 | |a On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups |
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| 520 | |a Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. | ||
| 650 | 4 | |a Finite group |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a -Soluble group |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a -Nilpotent group |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a -Critical |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Schmidt group |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Guo, Wenbin |e verfasserin |4 aut | |
| 700 | 1 | |a Zhang, Chi |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Monatshefte für Mathematik |d Wien [u.a.] : Springer, 1890 |g 189(2018), 2 vom: 07. Juni, Seite 239-242 |w (DE-627)254638058 |w (DE-600)1462913-6 |x 1436-5081 |7 nnns |
| 773 | 1 | 8 | |g volume:189 |g year:2018 |g number:2 |g day:07 |g month:06 |g pages:239-242 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s00605-018-1201-z |z lizenzpflichtig |3 Volltext |
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10.1007/s00605-018-1201-z doi (DE-627)SPR007216033 (SPR)s00605-018-1201-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Cao, Chenchen verfasserin aut On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 Guo, Wenbin verfasserin aut Zhang, Chi verfasserin aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 189(2018), 2 vom: 07. Juni, Seite 239-242 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:189 year:2018 number:2 day:07 month:06 pages:239-242 https://dx.doi.org/10.1007/s00605-018-1201-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_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_267 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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_4393 GBV_ILN_4700 31.00 ASE AR 189 2018 2 07 06 239-242 |
| spelling |
10.1007/s00605-018-1201-z doi (DE-627)SPR007216033 (SPR)s00605-018-1201-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Cao, Chenchen verfasserin aut On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 Guo, Wenbin verfasserin aut Zhang, Chi verfasserin aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 189(2018), 2 vom: 07. Juni, Seite 239-242 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:189 year:2018 number:2 day:07 month:06 pages:239-242 https://dx.doi.org/10.1007/s00605-018-1201-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_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_267 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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_4393 GBV_ILN_4700 31.00 ASE AR 189 2018 2 07 06 239-242 |
| allfields_unstemmed |
10.1007/s00605-018-1201-z doi (DE-627)SPR007216033 (SPR)s00605-018-1201-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Cao, Chenchen verfasserin aut On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 Guo, Wenbin verfasserin aut Zhang, Chi verfasserin aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 189(2018), 2 vom: 07. Juni, Seite 239-242 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:189 year:2018 number:2 day:07 month:06 pages:239-242 https://dx.doi.org/10.1007/s00605-018-1201-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_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_267 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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_4393 GBV_ILN_4700 31.00 ASE AR 189 2018 2 07 06 239-242 |
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10.1007/s00605-018-1201-z doi (DE-627)SPR007216033 (SPR)s00605-018-1201-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Cao, Chenchen verfasserin aut On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 Guo, Wenbin verfasserin aut Zhang, Chi verfasserin aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 189(2018), 2 vom: 07. Juni, Seite 239-242 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:189 year:2018 number:2 day:07 month:06 pages:239-242 https://dx.doi.org/10.1007/s00605-018-1201-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_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_267 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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_4393 GBV_ILN_4700 31.00 ASE AR 189 2018 2 07 06 239-242 |
| allfieldsSound |
10.1007/s00605-018-1201-z doi (DE-627)SPR007216033 (SPR)s00605-018-1201-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Cao, Chenchen verfasserin aut On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 Guo, Wenbin verfasserin aut Zhang, Chi verfasserin aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 189(2018), 2 vom: 07. Juni, Seite 239-242 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:189 year:2018 number:2 day:07 month:06 pages:239-242 https://dx.doi.org/10.1007/s00605-018-1201-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_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_267 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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_4393 GBV_ILN_4700 31.00 ASE AR 189 2018 2 07 06 239-242 |
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English |
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Enthalten in Monatshefte für Mathematik 189(2018), 2 vom: 07. Juni, Seite 239-242 volume:189 year:2018 number:2 day:07 month:06 pages:239-242 |
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Enthalten in Monatshefte für Mathematik 189(2018), 2 vom: 07. Juni, Seite 239-242 volume:189 year:2018 number:2 day:07 month:06 pages:239-242 |
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findex.gbv.de |
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Finite group -Soluble group -Nilpotent group -Critical Schmidt group |
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Monatshefte für Mathematik |
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Cao, Chenchen @@aut@@ Guo, Wenbin @@aut@@ Zhang, Chi @@aut@@ |
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2018-06-07T00:00:00Z |
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G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Finite group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">-Soluble group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">-Nilpotent group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">-Critical</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Schmidt group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Guo, Wenbin</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Zhang, Chi</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Monatshefte für Mathematik</subfield><subfield code="d">Wien [u.a.] : Springer, 1890</subfield><subfield code="g">189(2018), 2 vom: 07. 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Cao, Chenchen |
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Cao, Chenchen ddc 510 bkl 31.00 misc Finite group misc -Soluble group misc -Nilpotent group misc -Critical misc Schmidt group On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups |
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510 ASE 31.00 bkl On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups Finite group (dpeaa)DE-He213 -Soluble group (dpeaa)DE-He213 -Nilpotent group (dpeaa)DE-He213 -Critical (dpeaa)DE-He213 Schmidt group (dpeaa)DE-He213 |
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ddc 510 bkl 31.00 misc Finite group misc -Soluble group misc -Nilpotent group misc -Critical misc Schmidt group |
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On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups |
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on the structure of %$\mathfrak {n}_{\sigma }%$-critical groups |
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On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups |
| abstract |
Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. |
| abstractGer |
Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. |
| abstract_unstemmed |
Abstract Let G be a finite group, %$\sigma =\{\sigma _{i}|i\in I\}%$ be a partition of the set of all primes %$\mathbb {P}%$ and %$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}%$. G is said to be %$\sigma %$-primary if %$|\sigma (G)|\le 1%$; %$\sigma %$-soluble if every chief factor of G is %$\sigma %$-primary. A set %$\mathcal {H}%$ of subgroups of G is said to be a complete Hall%$\sigma %$-set of G if every non-identity member of %$\mathcal {H}%$ is a Hall %$\sigma _i%$-subgroup of G for some %$\sigma _i%$ and %$\mathcal {H}%$ contains exactly one Hall %$\sigma _i%$-subgroup for every %$\sigma _i\in \sigma (G)%$. G is said to be %$\sigma %$-full if G possesses a complete Hall %$\sigma %$-set; %$\sigma %$-nilpotent if G has a complete Hall %$\sigma %$-set %$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}%$ such that %$G=H_1\times H_2\times \cdots \times H_t%$; %$\mathfrak {N}_{\sigma }%$-critical (resp. %$\mathfrak {N}%$-critical) if G is not %$\sigma %$-nilpotent (resp. nilpotent) but all proper subgroups of G are %$\sigma %$-nilpotent (resp. nilpotent). An %$\mathfrak {N}%$-critical group is also called a Schmidt group. In this paper, we first prove that every %$\mathfrak {N}_{\sigma }%$-critical group is %$\sigma %$-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that %$\mathfrak {N}_{\sigma }%$-critical groups are also Schmidt groups and so the structure of %$\mathfrak {N}_{\sigma }%$-critical groups is obtained. |
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| title_short |
On the structure of %$\mathfrak {N}_{\sigma }%$-critical groups |
| url |
https://dx.doi.org/10.1007/s00605-018-1201-z |
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| author2 |
Guo, Wenbin Zhang, Chi |
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| doi_str |
10.1007/s00605-018-1201-z |
| up_date |
2024-07-04T02:27:50.065Z |
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| score |
7.402916 |