Document Zbl 1477.20033

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Math. J.</a> <a href="/?q=in%3A456219" title="Articles in this Issue">63, No. 1, 121-132 (2021)</a>. </div> <div class="abstract">In 2015, <span class="zbmathjax-textit">A. N. Skiba</span> [J. Algebra 436, 1–16 (2015; <a href="/1316.20020">Zbl 1316.20020</a>)] introduced the concepts of $\sigma$-primary, $\sigma$-nilpotent, and $\sigma$-soluble groups. Let $G^{\mathfrak{N}\sigma}$ be the $\sigma$-nilpotent residual of a finite group $G$, that is, the intersection of all normal subgroups $N$ of $G$ with $\sigma$-nilpotent quotient $G/N$. If $G$ is $\sigma$-soluble, then the $\sigma$-nilpotent length $($denoted by $l_{\sigma}(G))$ of $G$ is the length of the shortest normal chain of $G$ with $\sigma$-nilpotent factors. In this article, the authors introduce the $\sigma$-nilpotent norm of $G$. Let $N_{\sigma}(G)$ be the intersection of the normalizers of the $\sigma$-nilpotent residuals of all subgroups of $G$. Then the subgroup $N_{\sigma}(G)$ is called the $\sigma$-nilpotent norm of $G$. The authors investigate the relationship of the $\sigma$-nilpotent length with the $\sigma$-nilpotent norm of $G$. In particular, the authors prove that the $\sigma$-nilpotent length of a $\sigma$-soluble group $G$ is at most $r$ $(r>1)$ if and only if $l_{\sigma}(G/N_{\sigma}(G))\leq r$.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A17144">Haoran Yu (Changchun)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7286317">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D10" title="MSC2020">20D10</a> </td> <td class="space"> Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D15" title="MSC2020">20D15</a> </td> <td class="space"> Finite nilpotent groups, $p$-groups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D30" title="MSC2020">20D30</a> </td> <td class="space"> Series and lattices of subgroups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20D20" title="MSC2020">20D20</a> </td> <td class="space"> Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3A%24%5Csigma%24-nilpotent+residual">$\sigma$-nilpotent residual</a>; <a href="/?q=ut%3A%24%5Csigma%24-nilpotent+length">$\sigma$-nilpotent length</a>; <a href="/?q=ut%3A%24%5Csigma%24-nilpotent+norm">$\sigma$-nilpotent norm</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1316.20020">Zbl 1316.20020</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1477.20033" data-ciurl="/ci/07286317" data-biburl="/bibtex/07286317.bib" data-amsurl="/amsrefs/07286317.bib" data-xmlurl="/xml/07286317.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07286317.pdf" title="Zbl 1477.20033 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1017/S0017089520000051" aria-label="DOI for “On the $\sigma$-nilpotent norm and the $\sigma$-nilpotent length of a finite group”" title="10.1017/S0017089520000051">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Robinson, D. J. S., A Course in the theory of groups (Springer-Verlag, New York, Heidelberg, Berlin, 1982). · <a href="/0483.20001" class="nowrap">Zbl 0483.20001</a></td> </tr><tr> <td>[2]</td> <td class="space">Skiba, A. N., On σ-subnormal and σ-permutable subgroups of finite groups, J. Algebra, 436 (2015), 1-16. · <a href="/1316.20020" class="nowrap">Zbl 1316.20020</a></td> </tr><tr> <td>[3]</td> <td class="space">Guo, W. and Skiba, A. N., Finite groups whose n-maximal subgroups are σ-subnormal, Sci. China Math.62(7) (2019), 1355-1372. · <a href="/1476.20020" class="nowrap">Zbl 1476.20020</a></td> </tr><tr> <td>[4]</td> <td class="space">Skiba, A. N., Some characterizations of finite σ-soluble PσT-groups, J. Alg. 495(1) (2018), 114-129. · <a href="/1378.20027" class="nowrap">Zbl 1378.20027</a></td> </tr><tr> <td>[5]</td> <td class="space">Doerk, K. and Hawkes, T., Finite soluble groups (Walter de Gruyter, Berlin, New York, 1992). · <a href="/0753.20001" class="nowrap">Zbl 0753.20001</a></td> </tr><tr> <td>[6]</td> <td class="space">Chunikhin, S. A., Subgroups of finite groups (Nauka i Tehnika, Minsk, 1964). · <a href="/0119.26704" class="nowrap">Zbl 0119.26704</a></td> </tr><tr> <td>[7]</td> <td class="space">Guo, W. and Skiba, A. N., On σ-supersoluble groups and one generalization of CLT-groups, J. Alg. 512(1) (2018), 92-108. · <a href="/1439.20013" class="nowrap">Zbl 1439.20013</a></td> </tr><tr> <td>[8]</td> <td class="space">Baer, R., Kern, Der, eine charkteristishe Untergruppe, Compos. Math.1 (1935), 254-283.</td> </tr><tr> <td>[9]</td> <td class="space">Baer, R., Norm and hypernorm, Publ. Math. Debrecen, 4 (1956), 347-350. · <a href="/0071.25302" class="nowrap">Zbl 0071.25302</a></td> </tr><tr> <td>[10]</td> <td class="space">Shen, Z., Shi, W. and Qian, G., On the norm of the nilpotent residuals of all subgroups of a finite group, J. Alg.352 (2012), 290-298. · <a href="/1255.20019" class="nowrap">Zbl 1255.20019</a></td> </tr><tr> <td>[11]</td> <td class="space">Selkin, V. M., On the π-decomposable norm of a finite group, Proceedings of Francisk Skorina Gomel State University103(4) (2018), 51-55. · <a href="/1432.20017" class="nowrap">Zbl 1432.20017</a></td> </tr><tr> <td>[12]</td> <td class="space">Selkin, V. M. and Kosenok, N. S., On the generalized norm of a finite group, Problems Phys. Math. Tech.34(4) (2018), 69-74.</td> </tr><tr> <td>[13]</td> <td class="space">Ballester-Bolinches, A. and Ezquerro, L. M., Classes of finite groups (Springer, Dordrecht, 2006). · <a href="/1102.20016" class="nowrap">Zbl 1102.20016</a></td> </tr><tr> <td>[14]</td> <td class="space">Hu, B., Huang, J. and Skiba, A. N., Characterizations of finite σ-nilpotent and σ-quasinilpotent groups, Bull. Malays. Math.Soc.42(5) (2019), 2091-2104. · <a href="/1480.20053" class="nowrap">Zbl 1480.20053</a></td> </tr><tr> <td>[15]</td> <td class="space">Su, N. and Wang, Y., On the normalizers of $\mathfrak{F} $ -residuals of all subgroups of a finite group, J. Alg. 392 (2013), 185-198. · <a href="/1312.20015" class="nowrap">Zbl 1312.20015</a></td> </tr><tr> <td>[16]</td> <td class="space">Belonogov, V. A., Finite groups all of whose 2-maximal subgroups are π-decomposable, Trudi Instituta Matematiki i Mekhaniki Uro RAN20(2) (2014), 29-43.</td> </tr><tr> <td>[17]</td> <td class="space">Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967). · <a href="/0217.07201" class="nowrap">Zbl 0217.07201</a></td> </tr><tr> <td>[18]</td> <td class="space">Gorenstein, D., Finite groups (Harper & Row Publishers, New York, Evanston, London, 1968). · <a href="/0185.05701" class="nowrap">Zbl 0185.05701</a></td> </tr><tr> <td>[19]</td> <td class="space">Shemetkov, L.A., Formations of finite groups (Nauka, Moscow, 1978). · <a href="/0496.20014" class="nowrap">Zbl 0496.20014</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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