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...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Zhang, L. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Schlagwörter: |
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| Anmerkung: |
© Pleiades Publishing, Ltd. 2017 |
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| Übergeordnetes Werk: |
Enthalten in: Siberian mathematical journal - Pleiades Publishing, 1966, 58(2017), 5 vom: Sept., Seite 915-922 |
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| Übergeordnetes Werk: |
volume:58 ; year:2017 ; number:5 ; month:09 ; pages:915-922 |
| Links: |
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| DOI / URN: |
10.1134/S0037446617050196 |
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| Katalog-ID: |
OLC203339137X |
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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). | ||
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| 650 | 4 | |a rank of a soluble group | |
| 650 | 4 | |a -quasinormal subgroup | |
| 650 | 4 | |a -maximal subgroup | |
| 650 | 4 | |a -soluble group | |
| 700 | 1 | |a Guo, W. |4 aut | |
| 700 | 1 | |a Skiba, A. N. |4 aut | |
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10.1134/S0037446617050196 doi (DE-627)OLC203339137X (DE-He213)S0037446617050196-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Pleiades Publishing, Ltd. 2017 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 rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group Guo, W. aut Skiba, A. N. aut Enthalten in Siberian mathematical journal Pleiades Publishing, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)129553573 (DE-600)220062-4 (DE-576)015009602 0037-4466 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://doi.org/10.1134/S0037446617050196 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 58 2017 5 09 915-922 |
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10.1134/S0037446617050196 doi (DE-627)OLC203339137X (DE-He213)S0037446617050196-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Pleiades Publishing, Ltd. 2017 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 rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group Guo, W. aut Skiba, A. N. aut Enthalten in Siberian mathematical journal Pleiades Publishing, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)129553573 (DE-600)220062-4 (DE-576)015009602 0037-4466 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://doi.org/10.1134/S0037446617050196 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 58 2017 5 09 915-922 |
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10.1134/S0037446617050196 doi (DE-627)OLC203339137X (DE-He213)S0037446617050196-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Pleiades Publishing, Ltd. 2017 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 rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group Guo, W. aut Skiba, A. N. aut Enthalten in Siberian mathematical journal Pleiades Publishing, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)129553573 (DE-600)220062-4 (DE-576)015009602 0037-4466 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://doi.org/10.1134/S0037446617050196 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 58 2017 5 09 915-922 |
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10.1134/S0037446617050196 doi (DE-627)OLC203339137X (DE-He213)S0037446617050196-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Pleiades Publishing, Ltd. 2017 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 rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group Guo, W. aut Skiba, A. N. aut Enthalten in Siberian mathematical journal Pleiades Publishing, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)129553573 (DE-600)220062-4 (DE-576)015009602 0037-4466 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://doi.org/10.1134/S0037446617050196 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 58 2017 5 09 915-922 |
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10.1134/S0037446617050196 doi (DE-627)OLC203339137X (DE-He213)S0037446617050196-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Zhang, L. verfasserin aut Some notes on the rank of a finite soluble group 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Pleiades Publishing, Ltd. 2017 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 rank of a soluble group -quasinormal subgroup -maximal subgroup -soluble group Guo, W. aut Skiba, A. N. aut Enthalten in Siberian mathematical journal Pleiades Publishing, 1966 58(2017), 5 vom: Sept., Seite 915-922 (DE-627)129553573 (DE-600)220062-4 (DE-576)015009602 0037-4466 nnns volume:58 year:2017 number:5 month:09 pages:915-922 https://doi.org/10.1134/S0037446617050196 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 58 2017 5 09 915-922 |
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Siberian mathematical journal |
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Some notes on the rank of a finite soluble group |
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Some notes on the rank of a finite soluble group |
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Zhang, L. |
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Siberian mathematical journal |
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2017 |
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Zhang, L. Guo, W. Skiba, A. N. |
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Zhang, L. |
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10.1134/S0037446617050196 |
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510 |
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some notes on the rank of a finite soluble group |
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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). © Pleiades Publishing, Ltd. 2017 |
| 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). © Pleiades Publishing, Ltd. 2017 |
| 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). © Pleiades Publishing, Ltd. 2017 |
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Some notes on the rank of a finite soluble group |
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