Geodesic
On a generalization of Frobenius' theorem to infinite groups
Sbornik. Mathematics, Tome 29 (1976) no. 4, pp. 441-451 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper the following theorem is proved. Theorem. Suppose $G$ is a group, $H$ is a subgroup, and $a$ is an element of prime order $p\ne2$ in $H$ such that a) {\it$(G, H)$ is a Frobenius pair, i.e. $H\cap g^{-1}Hg=1$ for all $g\in G\setminus H$}; b) {\it for any $g\in G\setminus H$ the group $\langle a,g^{-1}ag\rangle$ is finite. Then $G = F_p\leftthreetimes H$, where $F_p$ is a periodic group containing no $p$-elements, and either $H$ possesses a unique involution or $H=N_G(\langle a\rangle)$.} Examples of periodic groups are given to show that the conditions $p\ne2$ and b) are essential restrictions in the theorem. It is proved that in the class of periodic biprimitively finite groups the existence in a group $G$ of a Frobenius pair $(G, H)$ already implies that $G=F_p\leftthreetimes H$ and $G$ admits a partition, i.e. $F^\#_p = F_p\setminus\{1\}=G\setminus\bigcup_{x\in G}H^x$. Bibliography: 14 titles. 
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     title = {On a~generalization of {Frobenius'} theorem to infinite groups},
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A. I. Sozutov; V. P. Shunkov. On a generalization of Frobenius' theorem to infinite groups. Sbornik. Mathematics, Tome 29 (1976) no. 4, pp. 441-451. http://geodesic.mathdoc.fr/item/SM_1976_29_4_a1/

[1] P. S. Novikov, S. I. Adyan, “O beskonechnykh periodicheskikh gruppakh”, Izv. AN SSSR, seriya matem., 32 (1968), 212–244 ; 251–524 ; 700–731 | Zbl

[2] S. I. Adyan, “O podgruppakh svobodnykh periodicheskikh grupp nechetnogo pokazatelya”, Trudy Matem. in-ta im. V. A. Steklova, CXII (1971), 64–72 | MR

[3] V. M. Busarkin, Yu. M. Gorchakov, Konechnye rasscheplyaemye gruppy, izd-vo «Nauka», Moskva, 1968 | MR

[4] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, izd-vo «Nauka», Moskva, 1972 | MR

[5] A. I. Starostin, “O gruppakh Frobeniusa”, Ukr. matem. zh., 23:5 (1971), 629–639 | MR | Zbl

[6] M. Kholl, Teoriya grupp, IL, Moskva, 1962

[7] S. N. Chernikov, “Gruppy s usloviem minimalnosti dlya neabelevykh podgrupp”, Gruppy s ogranicheniyami dlya podgrupp, Kiev, 1971, 96–105

[8] O. Yu. Shmidt, Lokalnaya konechnost odnogo klassa beskonechnykh periodicheskikh grupp, Izbr. trudy, izd-vo AN SSSR, Moskva, 1959

[9] V. P. Shunkov, “O nekotorom obobschenii teoremy Frobeniusa na periodicheskie gruppy”, Algebra i logika, 6:3 (1967), 113–124 | MR

[10] V. P. Shunkov, “Ob abelevykh podgruppakh v bioprimitivno konechnykh gruppakh”, Algebra i logika, 12:5 (1973), 603–614 | Zbl

[11] G. Frobenius, “Über auflösbare Gruppen, IV”, Sitzungsber. Preuss. Akad. Wiss. zu Berlin, 1901, 1216–1230 | Zbl

[12] W. Burnside, Theory of Groups of Finite Order, 1911

[13] M. Suzuki, “On finite groups with cyclic Sylow subroups for all odd primes”, Amer. J. Math., 77 (1955), 657–691 | DOI | MR | Zbl

[14] H. Zassenhaus, “Über endliche Fastkörper”, Hamb. Abh., 11 (1936), 187–220 | DOI