{"title":"所有非正则子群的核心都在中心的有限 p 群","authors":"Libo Zhao, Yangming Li, Lü Gong, Xiuyun Guo","doi":"10.1142/s0219498825502020","DOIUrl":null,"url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> be a finite <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-group. Then <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> is said to be a <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group if <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>H</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>≤</mo><mi>Z</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for every nonnormal subgroup <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>. In this paper, we study the <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> and get <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>≤</mo><mn>3</mn></math></span><span></span>. It is proved that <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">exp</mtext></mstyle><mo stretchy=\"false\">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">)</mo><mo>=</mo><mi>p</mi></math></span><span></span> if <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span><span></span> if <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>3</mn></math></span><span></span>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Finite p-groups in which the cores of all the nonnormal subgroups are in the center\",\"authors\":\"Libo Zhao, Yangming Li, Lü Gong, Xiuyun Guo\",\"doi\":\"10.1142/s0219498825502020\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span> be a finite <span><math altimg=\\\"eq-00004.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>p</mi></math></span><span></span>-group. Then <span><math altimg=\\\"eq-00005.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span> is said to be a <span><math altimg=\\\"eq-00006.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group if <span><math altimg=\\\"eq-00007.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><msub><mrow><mi>H</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>≤</mo><mi>Z</mi><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo stretchy=\\\"false\\\">)</mo></math></span><span></span> for every nonnormal subgroup <span><math altimg=\\\"eq-00008.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>H</mi></math></span><span></span> of <span><math altimg=\\\"eq-00009.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span>. In this paper, we study the <span><math altimg=\\\"eq-00010.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group <span><math altimg=\\\"eq-00011.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>G</mi></math></span><span></span> and get <span><math altimg=\\\"eq-00012.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>c</mi><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo stretchy=\\\"false\\\">)</mo><mo>≤</mo><mn>3</mn></math></span><span></span>. It is proved that <span><math altimg=\\\"eq-00013.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle><mtext mathvariant=\\\"normal\\\">exp</mtext></mstyle><mo stretchy=\\\"false\\\">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>′</mi></mrow></msup><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mi>p</mi></math></span><span></span> if <span><math altimg=\\\"eq-00014.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>c</mi><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\\\"eq-00015.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span><span></span> if <span><math altimg=\\\"eq-00016.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>c</mi><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mn>3</mn></math></span><span></span>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825502020\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
设 G 是有限 p 群。本文研究 CZ 群 G,得到 c(G)≤3。证明了当 c(G)=2 时,exp(G′)=p;当 c(G)=3 时,|G|≤p5。
Finite p-groups in which the cores of all the nonnormal subgroups are in the center
Let be a finite -group. Then is said to be a -group if for every nonnormal subgroup of . In this paper, we study the -group and get . It is proved that if and if .