Geodesic
Groups containing a~strongly embedded subgroup
Algebra i logika, Tome 48 (2009) no. 2, pp. 190-202.

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An involution $v$ of a group $G$ is said to be finite (in $G$) if $vv^g$ has finite order for any $v\in G$. A subgroup $B$ of $G$ is called a strongly embedded (in $G$) subgroup if $B$ and $G\setminus B$ contain involutions, but $B\cap B^g$ does not, for any $g\in G\setminus B$. We prove the following results. Theorem 1. Let a group $G$ contain a finite involution and an involution whose centralizer in $G$ is periodic. If every finite subgroup of $G$ of even order is contained in a simple subgroup isomorphic, for some $m$, to $L_2(2^m)$ or $Sz(2^m)$, then $G$ is isomorphic to $L_2(Q)$ or $Sz(Q)$ for some locally finite field $Q$ of characteristic two. In particular, $G$ is locally finite. Theorem 2. Let a group $G$ contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in $G$ is a 2-group, and every finite subgroup of even order in $G$ is contained in a finite non-Abelian simple subgroup of $G$, then $G$ is isomorphic to $L_2(Q)$ or $Sz(Q)$ for some locally finite field $Q$ of characteristic two.
Keywords: strongly embedded subgroup, involution, centralizer. 
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     title = {Groups containing a~strongly embedded subgroup},
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D. V. Lytkina; V. D. Mazurov. Groups containing a~strongly embedded subgroup. Algebra i logika, Tome 48 (2009) no. 2, pp. 190-202. http://geodesic.mathdoc.fr/item/AL_2009_48_2_a2/

[1] A. I. Sozutov, A. K. Shlëpkin, “O nekotorykh gruppakh s konechnoi involyutsiei, nasyschennykh konechnymi prostymi podgruppami”, Matem. zametki, 72:3 (2002), 433–447 | MR | Zbl

[2] K. A. Filippov, Gruppy, nasyschennye konechnymi neabelevymi prostymi gruppami i ikh tsentralnymi rasshireniyami, Dis. $\dots$ kand. fiz-mat. nauk, Krasnoyarsk, 2005

[3] B. Huppert, Endliche Gruppen, I, Grundlehren mathem. Wiss., 134, Springer-Verlag, Berlin etc., 1979 | MR | Zbl

[4] M. Suzuki, “On class of double transitive groups”, Ann. Math. (2), 75:1 (1962), 105–145 | DOI | MR | Zbl

[5] M. Suzuki, “On class of double transitive groups. II”, Ann. Math. (2), 79:3 (1964), 514–589 | DOI | MR | Zbl

[6] V. D. Mazurov, “O beskonechnykh gruppakh s abelevymi tsentralizatorami involyutsii”, Algebra i logika, 39:1 (2000), 74–86 | MR | Zbl

[7] B. Hartley, G. Shute, “Monomorphisms and direct limits of finite groups of Lie type”, Q. J. Math. Oxf. II Ser., 35:137 (1984), 49–71 | DOI | MR | Zbl

[8] V. P. Shunkov, $M_p$-gruppy, Nauka, M., 1990 | MR | Zbl

[9] D. V. Lytkina, V. D. Mazurov, “O gruppakh s silno vlozhennoi podgruppoi”, Matematicheskie sistemy, 7, Krasnoyarskii gos. agrarnyi un-t, Krasnoyarsk, 2009, 10–20