Geodesic
Tests for the nonsimplicity of factorable groups
Izvestiya. Mathematics , Tome 16 (1981) no. 2, pp. 261-278.

Voir la notice de l'article provenant de la source Math-Net.Ru

The following theorem is proved. Theorem. Suppose that a finite group $G$ is the product of two subgroups $A$ and $B,$ where $B$ is of odd order. Let at least one of the following conditions be satisfied: (a) $A$ is $2$-separable, and $(|A|,|B|)=1$. (b) $A$ is $2$-nilpotent with a $2$-separable derived group, $B$ is nilpotent, and $(|A|,|B|)=1$. (c) $A$ is supersolvable and $B$ is nilpotent. \noindent Then $O(A)$ lies in $O(G)$. Bibliography: 30 titles. 
@article{IM2_1981_16_2_a2,
     author = {L. S. Kazarin},
     title = {Tests for the nonsimplicity of factorable groups},
     journal = {Izvestiya. Mathematics },
     pages = {261--278},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {1981},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a2/}
}
TY  - JOUR
AU  - L. S. Kazarin
TI  - Tests for the nonsimplicity of factorable groups
JO  - Izvestiya. Mathematics 
PY  - 1981
SP  - 261
EP  - 278
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a2/
LA  - en
ID  - IM2_1981_16_2_a2
ER  - 
%0 Journal Article
%A L. S. Kazarin
%T Tests for the nonsimplicity of factorable groups
%J Izvestiya. Mathematics 
%D 1981
%P 261-278
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a2/
%G en
%F IM2_1981_16_2_a2
L. S. Kazarin. Tests for the nonsimplicity of factorable groups. Izvestiya. Mathematics , Tome 16 (1981) no. 2, pp. 261-278. http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a2/

[1] Belonogov V. A., Fomin A. N., Matrichnye predstavleniya konechnykh grupp, Nauka, M., 1976

[2] Berkovich Ya. G., “O $p$-gruppakh konechnogo poryadka”, Sib. matem. zh., 9:6 (1968), 1284–1306 | MR | Zbl

[3] Berkovich Ya. G., “Obobschenie teorem Kartera i Vilanda”, Dokl. AN SSSR, 171:4 (1966), 770–773 | Zbl

[4] Busarkin V. M., Gorchakov Yu. M., Konechnye rasscheplyaemye gruppy, Nauka, M., 1968 | MR

[5] Busarkin V. M., Sporadicheskie gruppy (obzor), Novosibirsk, Krasnoyarsk, 1976 | Zbl

[6] Kondratev A. S., “Konechnye prostye neabelevy gruppy, silovaya 2-podgruppa kotorykh est rasshirenie abelevoi gruppy posredstvom gruppy ranga 1”, Algebra i logika, 14:3 (1975), 288–303 | MR

[7] Kondratev A. S., “Konechnye prostye gruppy s silovymi 2-podgruppami poryadka $2^7$”, Izv. AN SSSR. Ser. matem., 41 (1977), 752–767

[8] Kourovskaya tetrad, izd-e 5, Novosibirsk, 1976

[9] Mazurov V. D., “Konechnye gruppy”, Itogi nauki. Algebra. Topologiya. Geometriya, 14, VINITI, M., 1976, 5–56 | MR | Zbl

[10] Fomin A. N., “Odno zamechanie o faktorizuemykh gruppakh”, Algebra i logika, 11:6 (1972), 608–611 | Zbl

[11] Chunikhin S. A., “O suschestvovanii podgrupp u konechnoi gruppy”, Trudy seminara po teorii grupp, M., L., 1938, 106–125

[12] Chunikhin S. A., Shemetkov L. A., “Konechnye gruppy”, Itogi nauki. Algebra. Topologiya. Geometriya, VINITI, M., 1971, 7–70 | Zbl

[13] Alperin J. L., Brauer R., Gorenstein D., “Finite groups with quasidihedral and wreathed Sylow 2-subgroups”, Trans. Amer. Math. Soc., 151:1 (1970), 1–261 | DOI | MR | Zbl

[14] Bender H., “Transitive Gruppen gerader Ordnaing in denen jede Involution genau einen Punkt festlasst”, J. Algebra, 17:4 (1971), 527–554 | DOI | MR | Zbl

[15] Chabot P., “Groups whose Sylow 2-subgroups have cyclic commutator groups. I, II, III”, J. Algebra, 19:1 (1971), 21–30 ; 21:2 (1972), 312–320 ; 29:3 (1974), 455–458 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[16] Curtis C. W., Kantor W. M., Seitz G. M., “The 2-transitive permutation representations of the finite Chevalle groups”, Trans. Amer. Math. Soc., 218 (1976), 1–59 | DOI | MR | Zbl

[17] Feit W., Thompson J. G., “Solvability of groups of odd order”, Pacific J. Math., 13:3 (1963), 775–1029 | MR | Zbl

[18] Glauberman G., “Fixed point subgroups that contain centralizers of involutions”, Math. Z., 124:4 (1972), 353–360 | DOI | MR

[19] Glauberman G., “Weakly closed elements of Sylow subgroups”, Math. Z., 107:1 (1968), 1–20 | DOI | MR | Zbl

[20] Goldschmidt D., “2-fusion in finite groups”, Ann. Math., 99:1 (1974), 70–117 | DOI | MR | Zbl

[21] Gorenstein D., Finite Groups, Harper and Row, N. Y., 1968 | MR | Zbl

[22] Gorenstein D., “Finite groups the centralizers of whose involutions have normal 2-complements”, Canad. J. Math., 21:2 (1969), 335–357 | MR | Zbl

[23] Hall J., “Fusion and dihedral 2-subgroups”, J. Algebra, 40:1 (1976), 203–228 | DOI | MR | Zbl

[24] Hanes H., Olson K., Scott W. R., “Products of simple groups”, J. Algebra, 36 (1975), 167–184 | DOI | MR | Zbl

[25] Huppert B., Endliche Gruppen. I, Berlin, 1967 | MR | Zbl

[26] Kegel O., “Producte nilpotenter Gruppen”, Arch. Math., 12:2 (1961), 90–93 | DOI | MR | Zbl

[27] Landrock P., “Finite groups with quasisimple component of type $PSU(3,2^n)$ on elementary abelian form”, Ill J. Math., 19:2 (1975), 198–230 | MR | Zbl

[28] Rowley P., “The $\pi$-separability of certain factorizable groups”, Math. Z., 153:3 (1977), 219–228 | DOI | MR | Zbl

[29] Suzuki M., “Finite groups in which the centralizer of any element of order 2 is 2-closed”, Ann. Math., 82:2 (1965), 191–212 | DOI | MR | Zbl

[30] Janko Z., Thompson J. G., “On finite simple groups whose Sylow 2-subgroups have no elementary subgroups of order 8”, Math. Z., 113:8 (1970), 385–397 | DOI | MR | Zbl