The structure of a finite group having $t$ (normal, abnormal) subgroups with pairwise mutually coprime indices and $\mathfrak{F}$-hypercentrality constraints on pairwise intersections of these subgroups, where $\mathfrak{F}$ is a saturated formation, is described. As a consequence, new characterizations of $\Sigma_t$-closed, weakly $R_t$-closed, and $N_0$-closed formations of finite groups are established. The obtained results make a significant contribution to the solution of the question posed by W. Guo and A.N. Skiba in 2011.
Keywords: finite group, product of groups, saturated formation, $\mathfrak{F}$-hypercenter, $\mathfrak{F}$-central chief factor
Received May 2, 2024
Revised June 4, 2024
Accepted June 10, 2024
Funding Agency: This work was supported by the Belarusian Republican Foundation for Fundamental Research (BRFFR-Φ23PHΦ М, project no. Φ23PHΦM-63).
Sergej Vladimirovich Balychev, Francisk Skorina Gomel State University, Gomel, 246019 Belarus, e-mail: sergej.balychev@gmail.com
Viachaslau Igaravich Murashka, Cand. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, Gomel, 246019 Belarus, e-mail: mvimath@yandex.ru
REFERENCES
1. Wielandt H. Über die normalstrukture von mehrfach faktorisierbaren Gruppen. J. Australian Math. Soc., 1960, vol. 1, no. 2, pp. 143–146. doi: 10.1017/S1446788700025520
2. Kegel O.H. Zur Struktur mehrafach faktorisierbarer endlicher Gruppen. Math. Z., 1965, vol. 87, pp. 42–48. doi: 10.1007/BF01109929
3. Doerk K. Minimal nicht überauflösbare, endlicher Gruppen. Math. Z., 1966, vol. 91, pp. 198–205. doi: 10.1007/BF01312426
4. Shemetkov L.A. Formatsii konechnykh grupp [Formations of finite groups]. Moscow, Nauka Publ., 1978, 271 p.
5. Kramer O.U. Endliche Gruppen mit Untergruppen mit paarweise teilerfremden Indizes. Math. Z., 1974, vol. 139, pp. 63–68. doi: 10.1007/BF01194145
6. Guo W., Skiba A.N. On factorizations of finite groups with \mathfrak{F}$-hypercentral intersections of the factors. J. Group Theory, 2011, vol. 14, pp. 695–708.
7. Guo W. Structure theory for canonical classes of finite groups. Berlin, Heidelberg, Springer-Verlag, 2015, 359 p. doi: 10.1007/978-3-662-45747-4
8. Kamornikov S.F., Sel’kin M.V. Podgruppovyye funktory i klassy konechnykh grupp [Subgroup functors and classes of finite groups]. Minsk, Belaruskaya Navuka, 2003, 254 p.
9. Vasil’ev A.F. Formations and their recognition. Izv. Gomel’skogo gos. univ. im.F.Skoriny, 2007, vol. 2, no. 41, pp. 23–29.
10. Friesen D. Products of normal supersoluble subgroups. Proc. Amer. Math. Soc., 1971, vol. 30, no. 1, pp. 46–48.
11. Baer R. Classes of finite groups and their properties. Illinois J. Math., 1957, vol. 1, pp. 115–187. doi: 10.1215/IJM/1255379396
12. Vasiliev A.F., Simonenko D.N. Relatively radical local formations. Iz. Gomel’skogo gos. universiteta im. F.Skoriny, 2006, no. 5(38), pp. 19–25 (in Russia).
13. Monakhov V.S., Chirik I.K. On the p-supersolvability of a finite factorizable group with normal factors. Trudy Inst. Math. Mekh. UrO RAN, 2015, vol. 21, no. 3, pp. 256–267 (in Russian).
14. Doerk K., Hawkes T. Finite soluble groups. Ser. De Gruyter expositions in mathematics, vol. 4, Berlin, NY, Walter de Gruyter Publ., 1992, 893 p.
15. Murashka V.I. On products of F*(G)-subnormal subgroups of finite groups. Math. Notes, 2022, vol. 111, no. 1–2, pp. 273–280. doi: 10.1134/S0001434622010308
Cite this article as: S.V. Balychev, V.I. Murashka. On the influence of the $\mathfrak{F}$-hypercenter on the structure of finite multifactorized groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 55–63.
