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Algebra i logika, 2018, Volume 57, Number 3, Pages 261–278
DOI: https://doi.org/10.17377/alglog.2018.57.301
(Mi al848)
 

This article is cited in 5 scientific papers (total in 5 papers)

Conjugacy of maximal and submaximal $\mathfrak X$-subgroups

W. Guoa, D. O. Revinbca

a Dep. Math., Univ. Sci. Tech. China, Hefei, 230026 P.R. China
b Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
c Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090 Russia
Full-text PDF (202 kB) Citations (5)
References:
Abstract: Let $\mathfrak X$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup $H$ of a finite group $G$ a submaximal $\mathfrak X$-subgroup if there exists an isomorpic embedding $\phi\colon G\hookrightarrow G^*$ of the group $G$ into some finite group $G^*$ under which $G^\phi$ is subnormal in $G^*$ and $H^\phi=K\cap G^\phi$ for some maximal $\mathfrak X$-subgroup $K$ of $G^*$. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal $\mathfrak X$-subgroups are conjugate in a finite group $G$ in which all maximal $\mathfrak X$-subgroups are conjugate? This question strengthens Wielandt's known problem of closedness for the class of $\mathscr D_\pi$-groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where $G$ is a simple group.
Keywords: finite group, maximal $\mathfrak X$-subgroup, submaximal $\mathfrak X$-subgroup, Hall $\pi$-subgroup, $\mathscr D_\pi$-property, $\mathscr D_\mathfrak X$-property.
Funding agency Grant number
National Natural Science Foundation of China 11771409
Chinese Academy of Sciences 2016VMA078
Russian Academy of Sciences - Federal Agency for Scientific Organizations I.1.1, 0314-2016-0001
Supported by the NNSF of China, grant No. 11771409.
Supported by Chinese Academy of Sciences President’s International Fellowship Initiative (grant No. 2016VMA078) and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2016-0001).
Received: 25.04.2017
Revised: 06.12.2017
English version:
Algebra and Logic, 2018, Volume 57, Issue 3, Pages 169–181
DOI: https://doi.org/10.1007/s10469-018-9490-9
Bibliographic databases:
Document Type: Article
UDC: 512.542.6
Language: Russian
Citation: W. Guo, D. O. Revin, “Conjugacy of maximal and submaximal $\mathfrak X$-subgroups”, Algebra Logika, 57:3 (2018), 261–278; Algebra and Logic, 57:3 (2018), 169–181
Citation in format AMSBIB
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\by W.~Guo, D.~O.~Revin
\paper Conjugacy of maximal and submaximal $\mathfrak X$-subgroups
\jour Algebra Logika
\yr 2018
\vol 57
\issue 3
\pages 261--278
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\crossref{https://doi.org/10.17377/alglog.2018.57.301}
\transl
\jour Algebra and Logic
\yr 2018
\vol 57
\issue 3
\pages 169--181
\crossref{https://doi.org/10.1007/s10469-018-9490-9}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и логика Algebra and Logic
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