×

A criterion for a finite group to be \(\sigma\)-soluble. (English) Zbl 1417.20003

Summary: Let \( \sigma= \{ \sigma_i | i\in I\} \) be a partition of the set of all primes \( \mathbb{P} \) and \(G\) a finite group. \(G\) is said to be \(\sigma\)-soluble if every chief factor \(H/K\) of \(G\) is \(\sigma\)-primary (that is, \(H/K\) is at \(\sigma_i\)-group for some \(i=i(H/K)\)). A subgroup \(A\) of \(G\) is called \(\sigma\)-subnormal in \(G\) if there is a subgroup chain \(A= A_0\leq A_1 \leq \cdots \leq A_n=G\) such that either \(A_{i-1} \trianglelefteq A_i\) or \(A_i / (A_{i-1})_{A_i}\) is \(\sigma\)-primary for all \(i=1,\dots,n\). Denote by \(i_\sigma (G)\) the number of classes of iso-ordic non-\(\sigma\)-subnormal subgroups of \(G\). In this note, we study the structure of \(G\) depending on the invariant \(i_\sigma (G)\). In particular, the following criterion is proved.
Theorem 1.2. If \(i_\sigma (G)\leq 2 |\sigma (G)| \) then \(G\) is \(\sigma\)-soluble.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
20D30 Series and lattices of subgroups
20D35 Subnormal subgroups of abstract finite groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Skiba, A. N., On σ-subnormal and σ-permutable subgroups of finite groups, J. Algebra, 436, 1-16 (2015) · Zbl 1316.20020
[2] Kegel, O. H., Untergruppenverbande endlicher gruppen, die den subnormalteilerverband each enthalten, Arch. Math, 30, 1, 225-228 (1978) · Zbl 0943.20500
[3] Skiba, A. N., Some characterizations of finite σ-soluble \(####\)-groups, J. Algebra, 495, 114-129 (2018) · Zbl 1378.20027
[4] Skiba, A. N., A generalization of a Hall theorem, J. Algebra Appl, 15, 5, 1650085-1650036 (2016) · Zbl 1348.20021
[5] Beidleman, J. C.; Skiba, A. N., On \(####\)-quasinormal subgroups of finite groups, J. Group Theory, 20, 5, 955-964 (2017) · Zbl 1391.20015
[6] Guo, W.; Zhang, C.; Skiba, A. N.; Sinitsa, D. A., On \(####\)-permutably embedded subgroups of finite groups, Rend. del Seminario Mat. della Univ. di Padova, 139, 143-158 (2018) · Zbl 1483.20031 · doi:10.4171/RSMUP
[7] Huang, J.; Hu, B.; Wu, X., Finite groups all of whose subgroups are σ-subnormal or σ-abnormal, Commun. Algebra, 45, 10, 4542-4549 (2017) · Zbl 1378.20021
[8] Al-Sharo, K. A.; Skiba, A. N., On finite groups with σ-subnormal Schmidt subgroups, Commun. Algebra, 45, 10, 4158-4165 (2017) · Zbl 1384.20023
[9] Hu, B.; Huang, J.; Skiba, A. N., Groups with only σ-semipermutable and σ-abnormal subgroups, Acta. Math. Hungar, 153, 1, 236-248 (2017) · Zbl 1399.20013
[10] Guo, W.; Skiba, A. N., On the lattice of \(####\)-subnormal subgroups of a finite group, Bull. Aust. Math. Soc, 96, 2, 233-244 (2017) · Zbl 1433.20004
[11] Guo, W.; Skiba, A. N., On Π-quasinormal subgroups of finite groups, Monatsh Math, 185, 3, 443-453 (2018) · Zbl 1468.20031 · doi:10.1007/s00605-016-1007-9
[12] Skiba, A. N., On some results in the theory of finite partially soluble groups, Commun. Math. Stat, 4, 3, 281-309 (2016) · Zbl 1394.20011
[13] Lu, J.; Meng, W., Finite groups with non-subnormal subgroups, Commun. Algebra, 45, 5, 2043-2046 (2017) · Zbl 1369.20027
[14] Shemetkov, L. A., Formations of Finite Groups (1978), Moscow: Nauka, Main Editorial Board for Physical and Mathematical Literature, Moscow · Zbl 0496.20014
[15] Guo, W.; Skiba, A. N., Finite groups with permutable complete Wielandt sets of subgroups, J. Group Theory, 18, 191-200 (2015) · Zbl 1332.20020
[16] Chunikhin, S. A., Subgroups of Finite Groups (1964), Minsk: Nauka i Tehnika, Minsk · Zbl 0119.26704
[17] Huppert, B., Endliche Gruppen I (1967), Berlin; Heidelberg; New York: Springer-Verlag, Berlin; Heidelberg; New York
[18] Guo, W.; Skiba, A. N., Finite groups whose n-maximal subgroups are σ-subnormal, Sci China. Math · Zbl 1476.20020 · doi:10.1007/s11425-016-9211-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.