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Mat. Zametki, 2001, Volume 69, Issue 6, Pages 912–918 (Mi mz705)  

This article is cited in 2 scientific papers (total in 2 papers)

On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A. I. Sozutov


Abstract: A proper subgroup $H$ of a group $G$ is said to be strongly isolated if it contains the centralizer of any nonidentity element of $H$ and 2-isolated if the conditions $C_G(g)\cap H\ne1$ and $2\in\pi(C_G(g))$ imply that $C_G(g)\le H$. An involution $i$ in a group $G$ is said to be finite if $|ii^g|<\infty$ ($\forall g\in G$). In the paper we study a group $G$ with finite involution $i$ and with a 2-isolated locally finite subgroup $H$ containing an involution. It is proved that at least one of the following assertions holds:
  • 1) all 2-elements of the group $G$ belong to $H$;
  • 2) $(G,H)$ is a Frobenius pair, $H$ coincides with the centralizer of the only involution in $H$, and all involutions in $G$ are conjugate;
  • 3) $G=F\leftthreetimes C_G(i)$ is a locally finite Frobenius group with Abelian kernel $F$;
  • 4) $H=V\leftthreetimes D$ is a Frobenius group with locally cyclic noninvariant factor $D$ and a strongly isolated kernel $V$, $U=O_2(V)$ is a Sylow 2-subgroup of the group $G$, and $G$ is a $Z$-group of permutations of the set $\Omega=\{U^g\mid g\in G\}$.


DOI: https://doi.org/10.4213/mzm705

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English version:
Mathematical Notes, 2001, 69:6, 833–838

Bibliographic databases:

UDC: 512.544
Received: 05.04.2000

Citation: A. I. Sozutov, “On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period”, Mat. Zametki, 69:6 (2001), 912–918; Math. Notes, 69:6 (2001), 833–838

Citation in format AMSBIB
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\by A.~I.~Sozutov
\paper On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period
\jour Mat. Zametki
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\vol 69
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\pages 912--918
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\zmath{https://zbmath.org/?q=an:1029.20017}
\elib{http://elibrary.ru/item.asp?id=5022586}
\transl
\jour Math. Notes
\yr 2001
\vol 69
\issue 6
\pages 833--838
\crossref{https://doi.org/10.1023/A:1010290717481}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169913100024}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Senashov, A. I. Sozutov, V. P. Shunkov, “Investigation of groups with finiteness conditions in Krasnoyarsk”, Russian Math. Surveys, 60:5 (2005), 805–848  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Jabara E., “A Note on Groups Covered by Conjugates of a Proper Subgroup”, J. Algebra, 370 (2012), 171–175  crossref  mathscinet  zmath  isi  elib  scopus  scopus
  • Математические заметки Mathematical Notes
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