Algebra i logika
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra Logika: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (194 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 512.542

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} 

• http://mi.mathnet.ru/eng/al80
• http://mi.mathnet.ru/eng/al/v43/i4/p411

 SHARE:

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
2. A. Trofimuk, “Derived Length of Finite Groups with Restrictions on Sylow Subgroups”, Math. Notes, 87:2 (2010), 264–270
3. V. S. Monakhov, A. A. Trofimuk, “Invarianty konechnykh razreshimykh grupp”, PFMT, 2010, no. 1(2), 63–81
4. D. A. Khodanovich, “O proizvodnoi dline konechnoi razreshimoi gruppy”, PFMT, 2011, no. 4(9), 106–110
5. Viktor Monakhov, Alexander Trofimuk, “Invariants of finite solvable groups”, Algebra Discrete Math., 14:1 (2012), 107–131
6. Monakhov V., Sokhor I., “on Cofactors of Subnormal Subgroups”, J. Algebra. Appl., 15:9 (2016), 1650169
7. Monakhov V.S., Sokhor I.L., “On Groups With Formational Subnormal Sylow Subgroups”, J. Group Theory, 21:2 (2018), 273–287
8. V. N. Knyagina, V. S. Monakhov, “Finite groups with semi-subnormal Schmidt subgroups”, Algebra Discrete Math., 29:1 (2020), 66–73
9. V. N. Knyagina, “Konechnye gruppy s subnormalnymi kommutantami $B$-podgrupp”, PFMT, 2021, no. 3(48), 73–75
•  Number of views: This page: 344 Full text: 99 References: 54 First page: 1