Some notes on the rank of a finite soluble group

Let G be a finite group and let σ = {σi|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Ha...
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Autor*in:	Zhang, L. [verfasserIn] Guo, W. [verfasserIn] Skiba, A. N. [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2017

Schlagwörter:	finite group rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group

Übergeordnetes Werk:	Enthalten in: Siberian mathematical journal - New York, NY [u.a.] : Consultants Bureau, 1966, 58(2017), 5 vom: Sept., Seite 915-922
Übergeordnetes Werk:	volume:58 ; year:2017 ; number:5 ; month:09 ; pages:915-922

Links:	Volltext

DOI / URN:	10.1134/S0037446617050196

Katalog-ID:	SPR017691540

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520			\|a Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).
650		4	\|a finite group \|7 (dpeaa)DE-He213
650		4	\|a rank of a soluble group \|7 (dpeaa)DE-He213
650		4	\|a -quasinormal subgroup \|7 (dpeaa)DE-He213
650		4	\|a -maximal subgroup \|7 (dpeaa)DE-He213
650		4	\|a -soluble group \|7 (dpeaa)DE-He213
700	1		\|a Guo, W. \|e verfasserin \|4 aut
700	1		\|a Skiba, A. N. \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Siberian mathematical journal \|d New York, NY [u.a.] : Consultants Bureau, 1966 \|g 58(2017), 5 vom: Sept., Seite 915-922 \|w (DE-627)325572348 \|w (DE-600)2037555-4 \|x 1573-9260 \|7 nnns
773	1	8	\|g volume:58 \|g year:2017 \|g number:5 \|g month:09 \|g pages:915-922
856	4	0	\|u https://dx.doi.org/10.1134/S0037446617050196 \|z lizenzpflichtig \|3 Volltext
912			\|a GBV_USEFLAG_A
912			\|a SYSFLAG_A
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912			\|a GBV_ILN_20
912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_206
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
912			\|a GBV_ILN_2086
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4393
912			\|a GBV_ILN_4700
936	b	k	\|a 31.00 \|q ASE
951			\|a AR
952			\|d 58 \|j 2017 \|e 5 \|c 09 \|h 915-922

Indexfelder

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hierarchy_sort_str	2017
bklnumber	31.00
publishDate	2017
allfields	10.1134/S0037446617050196 doi (DE-627)SPR017691540 (SPR)S0037446617050196-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H). finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213 Guo, W. verfasserin aut Skiba, A. N. verfasserin aut Enthalten in Siberian mathematical journal New York, NY [u.a.] : Consultants Bureau, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)325572348 (DE-600)2037555-4 1573-9260 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://dx.doi.org/10.1134/S0037446617050196 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_206 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_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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 58 2017 5 09 915-922
spelling	10.1134/S0037446617050196 doi (DE-627)SPR017691540 (SPR)S0037446617050196-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H). finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213 Guo, W. verfasserin aut Skiba, A. N. verfasserin aut Enthalten in Siberian mathematical journal New York, NY [u.a.] : Consultants Bureau, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)325572348 (DE-600)2037555-4 1573-9260 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://dx.doi.org/10.1134/S0037446617050196 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_206 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_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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 58 2017 5 09 915-922
allfields_unstemmed	10.1134/S0037446617050196 doi (DE-627)SPR017691540 (SPR)S0037446617050196-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H). finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213 Guo, W. verfasserin aut Skiba, A. N. verfasserin aut Enthalten in Siberian mathematical journal New York, NY [u.a.] : Consultants Bureau, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)325572348 (DE-600)2037555-4 1573-9260 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://dx.doi.org/10.1134/S0037446617050196 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_206 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_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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 58 2017 5 09 915-922
allfieldsGer	10.1134/S0037446617050196 doi (DE-627)SPR017691540 (SPR)S0037446617050196-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H). finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213 Guo, W. verfasserin aut Skiba, A. N. verfasserin aut Enthalten in Siberian mathematical journal New York, NY [u.a.] : Consultants Bureau, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)325572348 (DE-600)2037555-4 1573-9260 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://dx.doi.org/10.1134/S0037446617050196 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_206 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_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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 58 2017 5 09 915-922
allfieldsSound	10.1134/S0037446617050196 doi (DE-627)SPR017691540 (SPR)S0037446617050196-e DE-627 ger DE-627 rakwb eng 510 ASE 31.00 bkl Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H). finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213 Guo, W. verfasserin aut Skiba, A. N. verfasserin aut Enthalten in Siberian mathematical journal New York, NY [u.a.] : Consultants Bureau, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)325572348 (DE-600)2037555-4 1573-9260 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://dx.doi.org/10.1134/S0037446617050196 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_206 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_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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 58 2017 5 09 915-922
language	English
source	Enthalten in Siberian mathematical journal 58(2017), 5 vom: Sept., Seite 915-922 volume:58 year:2017 number:5 month:09 pages:915-922
sourceStr	Enthalten in Siberian mathematical journal 58(2017), 5 vom: Sept., Seite 915-922 volume:58 year:2017 number:5 month:09 pages:915-922
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	finite group rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group
dewey-raw	510
isfreeaccess_bool	false
container_title	Siberian mathematical journal
authorswithroles_txt_mv	Zhang, L. @@aut@@ Guo, W. @@aut@@ Skiba, A. N. @@aut@@
publishDateDaySort_date	2017-09-01T00:00:00Z
hierarchy_top_id	325572348
dewey-sort	3510
id	SPR017691540
language_de	englisch
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Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).</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">rank of a soluble group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">-quasinormal subgroup</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">-maximal subgroup</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="700" ind1="1" ind2=" "><subfield code="a">Guo, W.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Skiba, A. 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author	Zhang, L.
spellingShingle	Zhang, L. ddc 510 bkl 31.00 misc finite group misc rank of a soluble group misc -quasinormal subgroup misc -maximal subgroup misc -soluble group Some notes on the rank of a finite soluble group
authorStr	Zhang, L.
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topic_title	510 ASE 31.00 bkl Some notes on the rank of a finite soluble group finite group (dpeaa)DE-He213 rank of a soluble group (dpeaa)DE-He213 -quasinormal subgroup (dpeaa)DE-He213 -maximal subgroup (dpeaa)DE-He213 -soluble group (dpeaa)DE-He213
topic	ddc 510 bkl 31.00 misc finite group misc rank of a soluble group misc -quasinormal subgroup misc -maximal subgroup misc -soluble group
topic_unstemmed	ddc 510 bkl 31.00 misc finite group misc rank of a soluble group misc -quasinormal subgroup misc -maximal subgroup misc -soluble group
topic_browse	ddc 510 bkl 31.00 misc finite group misc rank of a soluble group misc -quasinormal subgroup misc -maximal subgroup misc -soluble group
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title	Some notes on the rank of a finite soluble group
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title_full	Some notes on the rank of a finite soluble group
author_sort	Zhang, L.
journal	Siberian mathematical journal
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author_browse	Zhang, L. Guo, W. Skiba, A. N.
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author-letter	Zhang, L.
doi_str_mv	10.1134/S0037446617050196
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title_sort	some notes on the rank of a finite soluble group
title_auth	Some notes on the rank of a finite soluble group
abstract	Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).
abstractGer	Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).
abstract_unstemmed	Let G be a finite group and let σ = {σi\|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a %${\sigma _{{i_j}}}%$-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G. Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).
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title_short	Some notes on the rank of a finite soluble group
url	https://dx.doi.org/10.1134/S0037446617050196
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author2	Guo, W. Skiba, A. N.
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up_date	2024-07-03T14:32:46.413Z
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