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Algebra Logika, 2004, Volume 43, Number 4, Pages 411–424 (Mi al80)  

This article is cited in 9 scientific papers (total in 9 papers)

Indices of Maximal Subgroups of Finite Soluble Groups

V. S. Monakhov

Francisk Skorina Gomel State University

Abstract: We look at the structure of a soluble group $G$ depending on the value of a function $m(G)=\max\limits_{p\in\pi(G)}$, where $m_p(G)=\max\{\log_p|G:M|\mid M<_{\max}G, |G:M|=p^a\}$, $p\in \pi (G)$.
\medskip Theorem 1. {\it States that for a soluble group $G$, (1) $r(G/\Phi (G))=m(G)$; (2) $d(G/\Phi (G))\leqslant1+\rho(m(G))\leqslant3+m(G)$; (3) $l_p(G)\leqslant1+t$, where $2^{t-1}<m_p(G)\leqslant 2^t$.}
\medskip Here, $\Phi(G)$ is the Frattini subgroup of $G$, and $r(G)$, $d(G)$, and $l_p(G)$ are, respectively, the principal rank, the derived length, and the $p$-length of $G$. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group $GL(n,F)$ of degree $n$, where $F$ is a field, is denoted by $\rho(n)$. The function $m(G)$ allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely,
\medskip Theorem 2. {\it Maintains that for any natural $k$, every soluble group $G$ contains a subgroup $K$ possessing the following properties: (1) $m(K)\leqslant k$; (2) if $T$ and $H$ are subgroups of $G$ such that $K\leqslant T<_{\max}H\leqslant G$ then $|H:T|=p^t$ for some prime $p$ and for $t>k$. Moreover, every two subgroups of $G$ enjoying (1) and (2) are mutually conjugate.}

Keywords: finite soluble group, maximal subgroup

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English version:
Algebra and Logic, 2004, 43:4, 230–237

Bibliographic databases:

UDC: 512.542
Received: 20.10.2002

Citation: V. S. Monakhov, “Indices of Maximal Subgroups of Finite Soluble Groups”, Algebra Logika, 43:4 (2004), 411–424; Algebra and Logic, 43:4 (2004), 230–237

Citation in format AMSBIB
\Bibitem{Mon04}
\by V.~S.~Monakhov
\paper Indices of Maximal Subgroups of Finite Soluble Groups
\jour Algebra Logika
\yr 2004
\vol 43
\issue 4
\pages 411--424
\mathnet{http://mi.mathnet.ru/al80}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2105846}
\zmath{https://zbmath.org/?q=an:1079.20027}
\elib{https://elibrary.ru/item.asp?id=9127554}
\transl
\jour Algebra and Logic
\yr 2004
\vol 43
\issue 4
\pages 230--237
\crossref{https://doi.org/10.1023/B:ALLO.0000035114.00094.62}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42249094561}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. S. Monakhov, A. Trofimuk, “Finite solvable groups in which the Sylow $p$-subgroups are either bicyclic or of order $p^3$”, J. Math. Sci., 167:6 (2010), 810–816  mathnet  crossref  mathscinet
    2. A. Trofimuk, “Derived Length of Finite Groups with Restrictions on Sylow Subgroups”, Math. Notes, 87:2 (2010), 264–270  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. V. S. Monakhov, A. A. Trofimuk, “Invarianty konechnykh razreshimykh grupp”, PFMT, 2010, no. 1(2), 63–81  mathnet
    4. D. A. Khodanovich, “O proizvodnoi dline konechnoi razreshimoi gruppy”, PFMT, 2011, no. 4(9), 106–110  mathnet
    5. Viktor Monakhov, Alexander Trofimuk, “Invariants of finite solvable groups”, Algebra Discrete Math., 14:1 (2012), 107–131  mathnet  mathscinet  zmath
    6. Monakhov V., Sokhor I., “on Cofactors of Subnormal Subgroups”, J. Algebra. Appl., 15:9 (2016), 1650169  crossref  mathscinet  zmath  isi  scopus
    7. Monakhov V.S., Sokhor I.L., “On Groups With Formational Subnormal Sylow Subgroups”, J. Group Theory, 21:2 (2018), 273–287  crossref  mathscinet  zmath  isi  scopus
    8. V. N. Knyagina, V. S. Monakhov, “Finite groups with semi-subnormal Schmidt subgroups”, Algebra Discrete Math., 29:1 (2020), 66–73  mathnet  crossref
    9. V. N. Knyagina, “Konechnye gruppy s subnormalnymi kommutantami $B$-podgrupp”, PFMT, 2021, no. 3(48), 73–75  mathnet  crossref
  • Алгебра и логика Algebra and Logic
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