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

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

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
\Bibitem{Soz01} \by A.~I.~Sozutov \paper On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period \jour Mat. Zametki \yr 2001 \vol 69 \issue 6 \pages 912--918 \mathnet{http://mi.mathnet.ru/mz705} \crossref{https://doi.org/10.4213/mzm705} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1861573} \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} 

• http://mi.mathnet.ru/eng/mz705
• https://doi.org/10.4213/mzm705
• http://mi.mathnet.ru/eng/mz/v69/i6/p912

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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
2. Jabara E., “A Note on Groups Covered by Conjugates of a Proper Subgroup”, J. Algebra, 370 (2012), 171–175
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