Abstract
Let \(\sigma =\{\sigma _{i}\ |\ i\in I\}\) be some partition of the set \(\mathbb {P}\) of all primes and G a finite group. A set \({{{\mathcal {H}}}}\) of subgroups of G is said to be a complete Hall \(\sigma \) -set of G if every member \(\ne 1\) of \({{{\mathcal {H}}}}\) is a Hall \(\sigma _{i}\)-subgroup of G for some \(i\in I\) and \({{\mathcal {H}}}\) contains exactly one Hall \(\sigma _{i}\)-subgroup of G for every i such that \(\sigma _{i}\cap \pi (G)\ne \emptyset \). A subgroup A of G is said to be \({{\mathcal {H}}}\)-permutable if A permutes with all members of the complete Hall \(\sigma \)-set \({{{\mathcal {H}}}}\) of G. In this paper, we study the structure of G under the assuming that some subgroups of G are \({{\mathcal {H}}}\)-permutable.
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Research of the first author is supported by a NNSF grant of China (Grant # 11371335) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences. Research of the third author is supported by Chinese Academy of Sciences Visiting Professorship for Senior International Scientists (Grant No. 2010T2J12).
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Guo, W., Cao, C., Skiba, A.N. et al. Finite Groups with \({{{\mathcal {H}}}}\)-Permutable Subgroups. Commun. Math. Stat. 5, 83–92 (2017). https://doi.org/10.1007/s40304-017-0101-1
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DOI: https://doi.org/10.1007/s40304-017-0101-1
Keywords
- Finite group
- Hall subgroup
- Complete Hall \(\sigma \)-set
- \({{{\mathcal {H}}}}\)-permutable subgroup
- PST-group