Geodesic
On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period
Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 912-918.

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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 $\Omega=\{U^g\mid g\in G\}$. 
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     author = {A. I. Sozutov},
     title = {On {Groups} with {Finite} {Involution} and {Locally} {Finite} {2-Isolated} {Subgroup} of {Even} {Period}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {912--918},
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     number = {6},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_6_a10/}
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A. I. Sozutov. On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period. Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 912-918. http://geodesic.mathdoc.fr/item/MZM_2001_69_6_a10/

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