On the Norm of the Abelian <i>p</i>-Group-Residuals

• Main Authors: Baojun Li, Yu Han, Lü Gong, Tong Jiang
• Format: Article
• Language: English
• Published: MDPI AG 2021-04-01
• Series: Mathematics
• Subjects: finite group; abelian p-group residual; soluble group; normalizer
• Online Access: https://www.mdpi.com/2227-7390/9/8/842

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>.