On \(\mathrm{CAP}_{S^*_p}\)-subgroups of finite groups. (English) Zbl 1473.20027
The paper under review deals with the structure of finite groups some of whose subgroups satisfy certain embedding properties.
A new class of non-solvable groups \({S^{*}_p}\) is defined, consisting of groups \(G\) such that every chief factor \(A/B\) of \(G\) satisfies one of the following conditions: (1) \(A/B\) is a \(p\)-group; (2) \(A/B\) is a \(p'\)-group; (3) \(|A/B|_p = p\). (Here, for an integer \(n\) and a prime number \(p\), \(n_p\) denotes the largest power of \(p\) dividing \(n\).) The class \({S^{*}_p}\) is a saturated formation which contains all \(p\)-soluble groups, but also some non-soluble groups.
Moreover, a subgroup \(H\) of a finite group \(G\) is said to be a \(\mathrm{CAP}_{S^{*}_p}\)-subgroup if for any \(pd\)-chief factor \(A/B\) of \(G\) (i.e. any chief factor such that \(p\) divides \(|A/B|\)) it either holds \(HA=HB \) or \(|H \cap A : H \cap B|_p \leq p\).
In this paper, various characterizations for a finite group to belong to the class \({S^{*}_p}\) are obtained under the assumption that some of its maximal subgroups or second maximal subgroups are \(\mathrm{CAP}_{S^{*}_p}\)-subgroups.
A new class of non-solvable groups \({S^{*}_p}\) is defined, consisting of groups \(G\) such that every chief factor \(A/B\) of \(G\) satisfies one of the following conditions: (1) \(A/B\) is a \(p\)-group; (2) \(A/B\) is a \(p'\)-group; (3) \(|A/B|_p = p\). (Here, for an integer \(n\) and a prime number \(p\), \(n_p\) denotes the largest power of \(p\) dividing \(n\).) The class \({S^{*}_p}\) is a saturated formation which contains all \(p\)-soluble groups, but also some non-soluble groups.
Moreover, a subgroup \(H\) of a finite group \(G\) is said to be a \(\mathrm{CAP}_{S^{*}_p}\)-subgroup if for any \(pd\)-chief factor \(A/B\) of \(G\) (i.e. any chief factor such that \(p\) divides \(|A/B|\)) it either holds \(HA=HB \) or \(|H \cap A : H \cap B|_p \leq p\).
In this paper, various characterizations for a finite group to belong to the class \({S^{*}_p}\) are obtained under the assumption that some of its maximal subgroups or second maximal subgroups are \(\mathrm{CAP}_{S^{*}_p}\)-subgroups.
Reviewer: Ana Martínez-Pastor (Valencia)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
finite groups; \(p\)-soluble groups; chief factors; \(\mathrm{CAP}_{S^{*}_p}\)-subgroups; maximal subgroupsReferences:
| [1] | Ballester-Bolinches, A.; Ezquerro, L. M., Classes of Finite Groups (2006), Springer · Zbl 1102.20016 |
| [2] | Ezquerro, L. M., A contribution to the theory of finite supersolvable groups, Rend. Sem. Mat. Univ. Padova, 89, 161-170 (1993) · Zbl 0809.20009 |
| [3] | Gillam, J. D., Cover-avoid subgroups in finite solvable groups, J. Algebra, 29, 2, 324-329 (1974) · Zbl 0293.20017 · doi:10.1016/0021-8693(74)90101-X |
| [4] | Guo, W., The Theory of Classes of Groups (2000), Beijing: Science Press-Kluwer Academic Publishers, Beijing |
| [5] | Guo, X.; Han, L.; Shum, K. P.; Shum, K. P.; Zelmanov, E.; Kolesniko, P.; Anita Wong, S. M., New Trends in Algebras and Combinatorics. Proceedings of the 3rd International Congress in Algebras and Combinatorics (ICAC 2017), On the cover-avoiding properties in finite groups, 88-104, World Scientific, Co. Pte. LtD · Zbl 1448.20023 |
| [6] | Guo, X.; Shum, K. P., Cover-avoidance properties and the structure of finite groups, J. Pure Appl. Algebra, 181, 2-3, 297-308 (2003) · Zbl 1028.20014 · doi:10.1016/S0022-4049(02)00327-4 |
| [7] | Guo, X.; Wang, J.; Shum, K. P., On semi-cover-avoiding maximal subgroups and solvability of finite groups, Commun. Algebra, 34, 9, 3235-3244 (2006) · Zbl 1106.20013 · doi:10.1080/00927870600778381 |
| [8] | Huppert, B.; Blackburn, N., Finite Groups III (1982), Berlin: Springer-Verlag, Berlin · Zbl 0514.20002 |
| [9] | Li, S.; Liu, H.; Liu, D., The solvability between finite groups and semi-subnormal-cover-avoidance subgroups, J. Math, 6, 1303-1308 (2017) · Zbl 1399.20016 |
| [10] | Liu, X.; Ding, N., On chief factors of finite groups, J. Pure Appl. Algebra, 210, 3, 789-796 (2007) · Zbl 1124.20011 · doi:10.1016/j.jpaa.2006.11.008 |
| [11] | Miao, L.; Lempken, W., On \(####\)-supplemented subgroups of finite groups, J. Group Theory, 12, 271-289 (2009) · Zbl 1202.20021 |
| [12] | Miao, L.; Zhang, J., On a class of non-solvable groups, J. Algebra, 496, 1-10 (2018) · Zbl 1421.20008 · doi:10.1016/j.jalgebra.2017.10.016 |
| [13] | Robinson, D. J. S., A Course in the Theory of Groups (1993), New York: Springer-Verlag, New York |
| [14] | Rose, J., On finite insolvable groups with nilpotent maximal subgroups, J. Algebra, 48, 1, 182-196 (1977) · Zbl 0364.20034 · doi:10.1016/0021-8693(77)90301-5 |
| [15] | Schaller, K. U., Über Deck-Meide-Untergruppen in endlichen auflösbaren Gruppen (1971) |
| [16] | Tomkinson, M. J., Cover-avoidance properties in finite solvable groups, Can. Math. Bull, 19, 2, 213-216 (1976) · Zbl 0352.20018 · doi:10.4153/CMB-1976-033-5 |
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