zbMATH — the first resource for mathematics

Groups all cyclic subgroups of which are BNA-subgroups. (English) Zbl 07030452
Ukr. Math. J. 69, No. 2, 331-336 (2017) and Ukr. Mat. Zh. 69, No. 2, 284-288 (2017).
Summary: Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\). We say that \(H\) is a BNA-subgroup of \(G\) if either \(H^x = H\) or \(x\in \langle H,H^x\rangle\) for all \(x\in G\). The BNA-subgroups of \(G\) are between normal and abnormal subgroups of \(G\). We obtain some new characterizations for finite groups based on the assumption that all cyclic subgroups are BNA-subgroups.
20 Group theory and generalizations
05 Combinatorics
Full Text: DOI
[1] B. Huppert, Endliche Gruppen I, Springer, Berlin, etc. (1967).
[2] Fattahi, A., Groups with only normal and abnormal subgroups, J. Algebra, 28, 15-19, (1974) · Zbl 0274.20022
[3] Ebert, G.; Bauman, S., A note on subnormal and abnormal chains, J. Algebra, 36, 287-293, (1975) · Zbl 0314.20019
[4] Cuccia, P.; Liotta, M., A condition on the minimal subgroups of a finite group, Boll. Unione Mat. Ital., 1, 303-308, (1982)
[5] Liu, J.; Li, S.; He, J., CLT-groups with normal or abnormal subgroups, J. Algebra, 362, 99-106, (2012) · Zbl 1261.20027
[6] X. He, S. Li, and Y. Wang, “On BNA-normality and solvability of finite groups,” Rend. Semin. Mat. Univ. Padova (2014); available online at http://rendiconti.math.unipd.it/forthcoming/downloads/HeLiWang\(−\)logo.pdf. · Zbl 1368.20010
[7] K. Doerk and T. O. Hawkes, Finite Soluble Groups, Walter De Gruyter, Berlin; New York (1992). · Zbl 0753.20001
[8] Li, S., On minimal subgroups of finite groups, Comm. Algebra, 22, 1913-1918, (1994) · Zbl 0799.20025
[9] Ballester-Bolinches, A.; Pedraza-Aguilera, MC, On minimal subgroup of finite groups, Acta Math. Hungar., 73, 335-342, (1996) · Zbl 0930.20021
[10] D. J. S. Robinson, A Course in the Theory of Groups, Springer, Berlin-New York (1993).
[11] Wei, H.; Wang, Y., On C⇤-normality and its properties, J. Group Theory, 10, 211-223, (2007) · Zbl 1125.20011
[12] M. S. Baand and Z. I. Borevich, “On the arrangement of intermediate subgroups,” in: Rings and Linear Groups, Kubanskii University, Krasnodar (1988), pp. 14-41.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.