Article
| Title: | Hypercyclically embedded property and supersolvable fusion systems (English) |
| Author: | Gao, Yaxin |
| Author: | Kaspczyk, Julian |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 541-563 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Given a prime $p$ and a subgroup $A$ of a finite group $G$, we say that $A$ is a \hbox {$p$-${\rm CAP}$-subgroup} of $G$ if $A$ covers or avoids every $p$-$G$-chief factor, where a $p$-$G$-chief factor is a $G$-chief factor of order divisible by $p$. We say that $A$ is a strong $p$-${\rm CAP}$-subgroup of $G$ if $A$ is a $p$-${\rm CAP}$-subgroup of any subgroup of $G$ containing $A$. We use the concept of strong $p$-${\rm CAP}$-subgroups to investigate the $p\mathfrak {F}$-hypercentrally embedded property of normal subgroups of a finite group and obtain some new results. Moreover, we extend the concept of (strong) $p$-${\rm CAP}$-subgroups to fusion systems and use this to characterize supersolvable and nilpotent fusion systems. (English) |
| Keyword: | finite group |
| Keyword: | hypercyclically embedded |
| Keyword: | subgroup |
| Keyword: | Sylow subgroup |
| Keyword: | fusion system |
| MSC: | 20D10 |
| MSC: | 20D20 |
| DOI: | 10.21136/CMJ.2026.0330-25 |
| . | |
| Date available: | 2026-05-22T11:21:44Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153648 |
| . | |
| Reference: | [1] Aschbacher, M., Kessar, R., Oliver, B.: Fusion Systems in Algebra and Topology.London Mathematical Society Lecture Note Series 391. Cambridge University Press, Cambridge (2011). Zbl 1255.20001, MR 2848834, 10.1017/CBO9781139003841 |
| Reference: | [2] Aseeri, F., Kaspczyk, J.: Criteria for supersolvability of saturated fusion systems.J. Algebra 647 (2024), 910-930. Zbl 1537.20038, MR 4722317, 10.1016/j.jalgebra.2024.02.023 |
| Reference: | [3] Ballester-Bolinches, A., Esteban-Romero, R., Meng, H., Su, N.: On finite $p$-groups of supersoluble type.J. Algebra 567 (2021), 1-10. Zbl 1506.20040, MR 4156098, 10.1016/j.jalgebra.2020.08.025 |
| Reference: | [4] Ballester-Bolinches, A., Ezquerro, L. M., Su, N., Wang, Y.: On the focal subgroup of a saturated fusion system.J. Algebra 468 (2016), 72-79. Zbl 1360.20015, MR 3550858, 10.1016/j.jalgebra.2016.07.034 |
| Reference: | [5] Chen, X., Guo, W.: On $\Pi$-supplemented subgroups of a finite group.Commun. Algebra 44 (2016), 731-745. Zbl 1339.20018, MR 3449950, 10.1080/00927872.2014.990019 |
| Reference: | [6] Chermak, A., Henke, E.: Fusion systems and localities -- A dictionary.Adv. Math. 410 (2022), Article ID 108690, 92 pages. Zbl 1514.20076, MR 4487972, 10.1016/j.aim.2022.108690 |
| Reference: | [7] Craven, D. A.: The Theory of Fusion Systems: An Algebraic Approach.Cambridge Studies in Advanced Mathematics 131. Cambridge University Press, Cambridge (2011). Zbl 1278.20001, MR 2808319, 10.1017/CBO9780511794506 |
| Reference: | [8] Doerk, K., Hawkes, T.: Finite Soluble Groups.De Gruyter Expositions in Mathematics 4. Walter de Gruyter, Berlin (1992). Zbl 0753.20001, MR 1169099, 10.1515/9783110870138 |
| Reference: | [9] Group, GAP: GAP: Groups, Algorithms, and Programming. Version 4.14.0.Available at https://www.gap-system.org/. |
| Reference: | [10] Glesser, A.: Sparse fusion systems.Proc. Edinb. Math. Soc., II. Ser. 56 (2013), 135-150. Zbl 1269.20013, MR 3021407, 10.1017/S0013091512000090 |
| Reference: | [11] Guo, W.: The Theory of Classes of Groups.Mathematics and its Applications (Dordrecht) 505. Kluwer, Dordrecht (2000). Zbl 1005.20016, MR 1862683, 10.1007/978-94-011-4054-6 |
| Reference: | [12] Guo, X., Shum, K. P.: Cover-avoidance properties and the structure of finite groups.J. Pure Appl. Algebra 181 (2003), 297-308. Zbl 1028.20014, MR 1975303, 10.1016/S0022-4049(02)00327-4 |
| Reference: | [13] Guo, W., Skiba, A. N.: Finite groups with generalized Ore supplement conditions for primary subgroups.J. Algebra 432 (2015), 205-227. Zbl 1329.20023, MR 3334146, 10.1016/j.jalgebra.2015.02.025 |
| Reference: | [14] He, X., Wang, Y.: On $p$-cover-avoid and $S$-quasinormally embedded subgroups in finite groups.J. Math. Res. Expo. 30 (2010), 743-750. Zbl 1224.20018, MR 2742129, 10.3770/j.issn:1000-341X.2010.04.019 |
| Reference: | [15] Henke, E.: Products in fusion systems.J. Algebra 376 (2013), 300-319. Zbl 1327.20018, MR 3003728, 10.1016/j.jalgebra.2012.11.037 |
| Reference: | [16] Huppert, B.: Endliche Gruppen. I.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin (1967), German. Zbl 0217.07201, MR 0224703, 10.1007/978-3-642-64981-3 |
| Reference: | [17] Kaspczyk, J.: On finite groups with given $IC\Phi$-subgroups.J. Algebra Appl. 23 (2024), Article ID 2450016, 13 pages. Zbl 1527.20016, MR 4688782, 10.1142/S0219498824500166 |
| Reference: | [18] Lei, D., Li, X., Guo, Y.: Weakly $s$-semipermutable subgroups and the $p\mathfrak{F}$-hypercenter of finite groups.Commun. Algebra 50 (2022), 4610-4618. Zbl 1514.20066, MR 4469913, 10.1080/00927872.2022.2069253 |
| Reference: | [19] Liao, J., Liu, Y.: Minimal non-nilpotent and locally nilpotent fusion systems.Algebra Colloq. 23 (2016), 455-462. Zbl 1359.20016, MR 3514534, 10.1142/S1005386716000432 |
| Reference: | [20] Liao, J., Zhang, J.: Nilpotent fusion systems.J. Algebra 442 (2015), 438-454. Zbl 1441.20014, MR 3395068, 10.1016/j.jalgebra.2015.03.002 |
| Reference: | [21] Ru, G., Zhang, S., Shen, Z.: Conditions for the supersolvability of $\mathcal{F}_S(G)$.Proc. Edinb. Math. Soc., II. Ser. 68 (2025), 566-572. Zbl 1569.20039, MR 4922253, 10.1017/S0013091524000865 |
| Reference: | [22] Shen, Z.: $p$-nilpotent fusion systems.J. Algebra Appl. 17 (2018), Article ID 1850235, 3 pages. Zbl 1406.20026, MR 3895207, 10.1142/S0219498818502353 |
| Reference: | [23] Shen, Z., Zhang, J.: $p$-supersolvable fusion systems.Sci. Sin. Math. 52 (2022), 1113-1120. 10.1360/SSM-2019-0320 |
| Reference: | [24] Su, N.: On supersolvable saturated fusion systems.Monatsh. Math. 187 (2018), 171-179. Zbl 1467.20014, MR 3842941, 10.1007/s00605-017-1145-8 |
| Reference: | [25] Su, N., Li, Y., Wang, Y.: A criterion of $p$-hypercyclically embedded subgroups of finite groups.J. Algebra 400 (2014), 82-93. Zbl 1300.20030, MR 3147365, 10.1016/j.jalgebra.2013.11.007 |
| Reference: | [26] Zhang, S., Shen, Z.: On generalized covering and avoidance properties of finite groups and saturated fusion systems.J. Algebra 666 (2025), 149-168. Zbl 1560.20061, MR 4837796, 10.1016/j.jalgebra.2024.11.030 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
