Abstract
A subgroup H of a group G is said to be G-semipermutable in G if H has a supplement T in G such that \(G=HT\) and for every subgroup \(T_1\) of T, \(HT_1^g=T_1^gH\) for some element g in G. In this paper, we investigate the structure of G under the assumption that a soluble maximal subgroup of G is G-semipermutable in G.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)
Guo, W.B.: Structure Theory for Canonical Classes of Finite groups. Springer, Heidelberg (2015)
Guo, W.B., Shum, K.P., Skiba, A.N.: \(X\)-semipermutable subgroups of finite groups. J. Algebra 315, 31–41 (2007)
Guralnick, R.T.: Subgroups of prime power index in a simple group. J. Algebra 81, 304–311 (1983)
Li, B.J., Skiba, A.N.: New characterizations of finite supersoluble groups. Sci. China Ser. A Math. 51(5), 827–841 (2008)
Li, J.B., Chen, G.Y.: The influence of \(X\)-semipermutability of subgroups on the structure of finite groups. Sci China Ser A Math. 52(2), 261–271 (2009)
Acknowledgements
The authors are much indebted to the reviewer whose valuable comments and detailed reports have greatly improved this manuscript. This work was partially supported by the NNSF of China (11501071, 11671063), the Scientific Research Foundation of CSTC (cstc2016jcyjA0065, cstc2017jcyjAX0329, cstc2018jcyjAX0060) and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1711273).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. S. Wilson.
Rights and permissions
About this article
Cite this article
Li, J., Yu, D. A characterization of solubility of finite groups. Monatsh Math 189, 691–694 (2019). https://doi.org/10.1007/s00605-018-1243-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-018-1243-2