{"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}
引用次数: 0
Abstract
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 .