On the Norm of the Abelian <i>p</i>-Group-Residuals
Let <i>G</i> be a group. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/8/842 |
| _version_ | 1797537907128401920 |
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| author | Baojun Li Yu Han Lü Gong Tong Jiang |
| author_facet | Baojun Li Yu Han Lü Gong Tong Jiang |
| author_sort | Baojun Li |
| collection | DOAJ |
| description | Let <i>G</i> be a group. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>⋂</mo><mrow><mi>H</mi><mo>≤</mo><mi>G</mi></mrow></msub><msub><mi>N</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>H</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined and, the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are investigated. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>=</mo><mi>D</mi><mo>(</mo><mi>P</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Sylow <i>p</i>-subgroup and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><mi>N</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a Hall <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mo>′</mo></msup></semantics></math></inline-formula>-subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, respectively. Furthermore, it is proved in a group <i>G</i> that (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>; (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>O</mi><msup><mi>p</mi><mo>′</mo></msup></msub><mrow><mo>(</mo><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≤</mo><msub><mi>Z</mi><mo>∞</mo></msub><mrow><mo>(</mo><msup><mi>O</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and (3) if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T12:22:55Z |
| format | Article |
| id | doaj.art-23f3845c979c4a49910061a10fa0f67d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:22:55Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-23f3845c979c4a49910061a10fa0f67d2023-11-21T15:18:03ZengMDPI AGMathematics2227-73902021-04-019884210.3390/math9080842On the Norm of the Abelian <i>p</i>-Group-ResidualsBaojun Li0Yu Han1Lü Gong2Tong Jiang3School of Sciences, Nantong University, Nantong 226019, ChinaSchool of Sciences, Nantong University, Nantong 226019, ChinaSchool of Sciences, Nantong University, Nantong 226019, ChinaSchool of Sciences, Nantong University, Nantong 226019, ChinaLet <i>G</i> be a group. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>⋂</mo><mrow><mi>H</mi><mo>≤</mo><mi>G</mi></mrow></msub><msub><mi>N</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>H</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined and, the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are investigated. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>=</mo><mi>D</mi><mo>(</mo><mi>P</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Sylow <i>p</i>-subgroup and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><mi>N</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a Hall <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mo>′</mo></msup></semantics></math></inline-formula>-subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, respectively. Furthermore, it is proved in a group <i>G</i> that (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>; (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>O</mi><msup><mi>p</mi><mo>′</mo></msup></msub><mrow><mo>(</mo><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≤</mo><msub><mi>Z</mi><mo>∞</mo></msub><mrow><mo>(</mo><msup><mi>O</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and (3) if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mrow><mo>(</mo><msup><mi>G</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>D</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/9/8/842finite groupabelian p-group residualsoluble groupnormalizer |
| spellingShingle | Baojun Li Yu Han Lü Gong Tong Jiang On the Norm of the Abelian <i>p</i>-Group-Residuals Mathematics finite group abelian p-group residual soluble group normalizer |
| title | On the Norm of the Abelian <i>p</i>-Group-Residuals |
| title_full | On the Norm of the Abelian <i>p</i>-Group-Residuals |
| title_fullStr | On the Norm of the Abelian <i>p</i>-Group-Residuals |
| title_full_unstemmed | On the Norm of the Abelian <i>p</i>-Group-Residuals |
| title_short | On the Norm of the Abelian <i>p</i>-Group-Residuals |
| title_sort | on the norm of the abelian i p i group residuals |
| topic | finite group abelian p-group residual soluble group normalizer |
| url | https://www.mdpi.com/2227-7390/9/8/842 |
| work_keys_str_mv | AT baojunli onthenormoftheabelianipigroupresiduals AT yuhan onthenormoftheabelianipigroupresiduals AT lugong onthenormoftheabelianipigroupresiduals AT tongjiang onthenormoftheabelianipigroupresiduals |