Abstract
Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ p and Σ p ∪Σ q , where q≠p, are G-covering subgroup systems for many classes of finite groups.
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Research of the first author is supported by a NNSF grant of China (Grant No. 11071229).
Research of the second author is supported by Chinese Academy of Sciences Visiting Professorship for Senior International Scientists (Grant No. 2010T2J12).
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Guo, W., Skiba, A.N. On G-covering subgroup systems of finite groups. Acta Math Hung 133, 376–386 (2011). https://doi.org/10.1007/s10474-011-0088-0
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DOI: https://doi.org/10.1007/s10474-011-0088-0
Key words and phrases
- supplement of subgroup
- G-covering subgroup system
- Sylow subgroup
- maximal subgroup
- p-supersoluble group
- p-nilpotent group
- p-decomposable group
- metanilpotent group