Finite groups with minimal weakly BNA-subgroups. (English) Zbl 07910524
Summary: Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is said to be a BNA-subgroup of \(G\) if either \(H^x = H\) or \(x \in \langle H, H^x \rangle\) for all \(x \in G\). A subgroup \(H\) of \(G\) is said to be a weakly BNA-subgroup of \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(G = HT\) and \(H \cap T\) is a BNA-subgroup of \(G\). In this paper, we investigate the structure of a finite group \(G\) under the assumption that every minimal subgroup of \(G\) not having a supersolvable supplement in \(G\) is a weakly BNA-subgroup of \(G\).
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
References:
| [1] | Doerk, K. and Hawkes, T., Finite Soluble Groups (Walter de Gruyter, New York, 1992). · Zbl 0753.20001 |
| [2] | Itô, N., Über eine zur Frattini-Gruppe duale Bildung, Nagoya Math. J.9 (1955) 123-127. · Zbl 0066.01401 |
| [3] | Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967). · Zbl 0217.07201 |
| [4] | Buckley, J., Finite groups whose minimal subgroups are normal, Math. Z.116(1) (1970) 15-17. · Zbl 0202.02303 |
| [5] | Yokoyama, A., Finite solvable groups whose \(\mathcal{F} \)-hypercenter contains all minimal subgroups, Arch. der Math.26(1) (1975) 123-130. · Zbl 0307.20012 |
| [6] | Yokoyama, A., Finite solvable groups whose \(\mathcal{F} \)-hypercenter contains all minimal subgroups II, Arch. der Math.27(1) (1976) 572-575. · Zbl 0348.20016 |
| [7] | Asaad, M., On the solvability of finite groups, Arch. der Math.51(4) (1988) 289-293. · Zbl 0656.20031 |
| [8] | Wang, Y., \(c\)-normality of groups and its properties, J. Algebra180(3) (1996) 954-965. · Zbl 0847.20010 |
| [9] | Li, D. and Guo, X., The influence of \(c\)-normality of subgroups on the structure of finite groups II, Commun. Algebra26(6) (1998) 1913-1922. · Zbl 0906.20012 |
| [10] | Li, D. and Guo, X., The influence of \(c\)-normality of subgroups on the structure of finite groups, J. Pure Appl. Algebra150(1) (2000) 53-60. · Zbl 0967.20011 |
| [11] | Wei, H., On \(c\)-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Commun. Algebra29(5) (2001) 2193-2200. · Zbl 0990.20012 |
| [12] | Wei, H., Wang, Y. and Li, Y., On \(c\)-Normal maximal and minimal subgroups of sylow subgroups of finite groups. II, Commun. Algebra31(10) (2003) 4807-4816. · Zbl 1050.20011 |
| [13] | He, X., Li, S. and Wang, Y., On BNA-normality and solvability of finite groups, Rendiconti Del Seminario Matematico Della Univ. Di Padova136 (2016) 51-60. · Zbl 1368.20010 |
| [14] | Guo, Q., He, X. and Liu, W., On weakly BNA-subgroups of finite groups, Commun. Algebra55(11) (2022) 4746-4755. · Zbl 1514.20064 |
| [15] | Shaalan, A., The influence of \(\pi \)-quasinormality of some subgroups on the structure of a finite group, Acta Math. Hungarica56(3) (1990) 287-293. · Zbl 0725.20018 |
| [16] | Skiba, A., On weakly \(s\)-permutable subgroups of finite groups, J. Algebra315(1) (2007) 192-209. · Zbl 1130.20019 |
| [17] | Ballester-Bolinches, A. and Pedraza-Aguilera, M., On minimal subgroups of finite groups, Acta Math. Hungarica73(4) (1996) 335-342. · Zbl 0930.20021 |
| [18] | Al-Shomrani, A., Ramadan, M. and Heliel, A., Finite groups whose minimal subgroups are weakly \(\mathcal{H} \)-subgroups, Acta Math. Sci.32(6) (2012) 2295-2301. · Zbl 1289.20034 |
| [19] | Weinstein, M., Between Nilpotent and Solvable (Polygonal Publishing House, Passaic, 1982). · Zbl 0488.20001 |
| [20] | Huppert, B. and Blackburn, N., Finite Groups III (Springer-Verlag, 1982). · Zbl 0514.20002 |
| [21] | He, X., Li, S. and Wang, Y., Groups all cyclic subgroups of which are BNA-Subgroups.Ukrainian Math. J.69(2) (2017) 331-336. · Zbl 1499.20037 |
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.
