Abstract
A subgroup H of a finite group G is said to satisfy the \({\Pi}\)-property in G if every prime dividing \({|G : {\rm N}_{G}(HK \cap L)|}\) also divides the order of \({(HK \cap L)/K}\), for every G-chief factor L/K of G. The \({\Pi}\)-property is a subgroup embedding property of arithmetical character introduced in Li (J. Algebra 334:321–337, 2011) that gathers common properties of many other well-known subgroup embeddings (most of them of normal type) and allows us to glimpse common behaviors to all of them. The aim of this note is to prove that a finite group G is soluble if and only if in G all maximal subgroups satisfy the \({\Pi}\)-property in G. This is the answer to a question posed by Li (J. Algebra 334:321–337, 2011, Question 5.2).
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Ballester-Bolinches, A., Jiménez-Seral, P., Li, X. et al. On the \({\Pi}\)-property of subgroups of finite groups. Arch. Math. 105, 301–305 (2015). https://doi.org/10.1007/s00013-015-0808-8
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DOI: https://doi.org/10.1007/s00013-015-0808-8