Geodesic
On the permutability of a Sylow subgroup with Schmidt subgroups from a supplement
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 145-154 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A Schmidt group is a finite nonnilpotent group each of whose proper subgroups is nilpotent. A supplement of a subgroup $A$ in a group $G$ is a subgroup $B$ of $G$ such that $G=AB$. Finite groups in which a Sylow subgroup is permutable with some Schmidt subgroups were studied by Ya.G. Berkovich and E.M. Pal'chik (Sib. Mat. Zh. 8(4), 741-753 (1967)) and by V. N. Knyagina and V.S. Monakhov (Proc. Steklov Inst. Math. 272 (Suppl. 1), S55-S64 (2011)). In this situation, the group may be nonsolvable. For example, in the group PSL(2,7) a Sylow 2-subgroup is permutable with all Shmidt subgroups of odd order. In the group SL(2,8) a Sylow 3-subgroup is permutable with all 2-closed Shmidt subgroups of even order. In the group SL(2,4) a Sylow 5-subgroup is permutable with every 2-closed Shmidt subgroup of even order. Since the groups Sz$(2^{2k+1})$ for $k\geq 1$, PSU(5,4), PSU(4,2), and PSp$(4,2^n)$ do not contain Shmidt subgroups of odd order, in these groups any Sylow subgroup is permutable with any Shmidt subgroup of odd order. We establish the $r$-solvability a finite group $G$ such that $r$ is odd and is not a Fermat prime and a Sylow $r$-subgroup $R$ is permutable with 2-nilpotent (or 2-closed) Schmidt subgroups of even order from some supplement of $R$ in $G$. We give examples showing that the constraints on $r$ are not superfluous.
Keywords: finite group, Schmidt group, Sylow r-subgroup.
Mots-clés : r-solvable group 
@article{TIMM_2018_24_3_a13,
     author = {V. S. Monakhov and E. V. Zubei},
     title = {On the permutability of a {Sylow} subgroup with {Schmidt} subgroups from a supplement},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {145--154},
     year = {2018},
     volume = {24},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2018_24_3_a13/}
}
TY  - JOUR
AU  - V. S. Monakhov
AU  - E. V. Zubei
TI  - On the permutability of a Sylow subgroup with Schmidt subgroups from a supplement
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2018
SP  - 145
EP  - 154
VL  - 24
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2018_24_3_a13/
LA  - ru
ID  - TIMM_2018_24_3_a13
ER  - 
%0 Journal Article
%A V. S. Monakhov
%A E. V. Zubei
%T On the permutability of a Sylow subgroup with Schmidt subgroups from a supplement
%J Trudy Instituta matematiki i mehaniki
%D 2018
%P 145-154
%V 24
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2018_24_3_a13/
%G ru
%F TIMM_2018_24_3_a13
V. S. Monakhov; E. V. Zubei. On the permutability of a Sylow subgroup with Schmidt subgroups from a supplement. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 145-154. http://geodesic.mathdoc.fr/item/TIMM_2018_24_3_a13/

[1] Shmidt O. Yu., “Gruppy, vse podgruppy kotorykh spetsialnye”, Mat. sb., 31 (1924), 366–372

[2] Monakhov V. S., “Podgruppy Shmidta, ikh suschestvovanie i nekotorye prilozheniya”, Cek. 1, Tr. Ukr. mat. kongressa (2001), In-t matematiki NAHU, Kiev, 2002, 81–90

[3] Berkovich Ya. G., Palchik E. M., “O perestanovochnosti podgrupp konechnoi gruppy”, Sib. mat. zhurn., 8:4 (1967), 741–753

[4] Knyagina V. N., Monakhov V. S., “O perestanovochnosti silovskikh podgrupp s podgruppami Shmidta”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16:3 (2010), 130–139

[5] Huppert B., Endliche Gruppen I, Springer, Berlin; Heidelberg; N Y, 1967, 793 pp. | DOI

[6] Monakhov V. S., Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysheishaya shkola, Minsk, 2006, 207 pp.

[7] Knyagina V. N., Monakhov V. S., “Konechnye gruppy s polunormalnymi podgruppami Shmidta”, Algebra i logika, 46:4 (2007), 448–458

[8] Su X., “On seminormal subgroups of finite group”, J. Math. (Wuhan), 8:1 (1988), 7–9

[9] Carocca A., Matos H., “Some solvability criteria for finite groups”, Hokkaido Math. J., 26:1 (1997), 157–161 | DOI

[10] Podgornaya V. V., “Polunormalnye podgruppy i sverkhrazreshimost konechnykh grupp”, Vestsi NAN Belarusi. Ser. fiz.-matem. navuk, 2000, no. 4, 22–25

[11] Monakhov V. S., “Konechnye gruppy s polunormalnoi khollovoi podgruppoi”, Mat. zam., 80:4 (2006), 573–581

[12] Berkovich Ya. G., “Teorema o nenilpotentnykh razreshimykh podgruppakh konechnoi gruppy”, Konechnye gruppy, Nauka i tekhnika, Minsk, 1966, 24–39

[13] Monakhov V. S., “O podgruppakh Shmidta konechnykh grupp”, Voprosy algebry, 1998, no. 13, 153–171

[14] Ito N., “On the faktorisations of the linear fractional group $LF(2,p^n)$”, Acta Sci. Math., 1953, no. 15, 79–84

[15] Monakhov V. S., “Proizvedenie konechnykh grupp, blizkikh k nilpotentnym”, Konechnye gruppy, Nauka i tekhnika, Minsk, 1975, 70–100

[16] Fisman E., “On the product of two finite solvable groups”, J. Algebra, 80 (1983), 517–536

[17] Kazarin L. S., “O gruppakh, predstavimykh v vide proizvedeniya dvukh razreshimykh podgrupp”, Commun. Algebra, 14:6 (1986), 1001–1066

[18] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp.

[19] Knyagina V. N., Monakhov V. S., “O $\pi^\prime$-svoistvakh konechnoi gruppy, obladayuschei $\pi$-khollovoi podgruppoi”, Sib. mat. zhurn., 52:2 (2011), 297–309