Abstract
We prove that \( G \) is a finite \( \sigma \)-soluble group with transitive \( \sigma \)-permutability if and only if the following hold: (i) \( G \) possesses a complete Hall \( \sigma \)-set \( {\mathcal{H}}=\{H_{1},\dots,H_{t}\} \) and a normal subgroup \( N \) with \( \sigma \)-nilpotent quotient \( G/N \) such that \( H_{i}\cap N\leq Z_{\mathfrak{U}}(H_{i}) \) for all \( i \); and (ii) every \( \sigma_{i} \)-subgroup of \( G \) is \( \tau_{\sigma} \)-permutable in \( G \) for all \( \sigma_{i}\in\sigma(N) \).
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Acknowledgment
The authors are very grateful for the helpful suggestions and remarks of the referee.
Funding
The authors were supported by the NNSF of China (11901364 and 11771409), the science and technology innovation project of colleges and universities in the Shanxi Province of China (2019L0747) and the applied basic research program project in the Shanxi Province of China (201901D211439).
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Mao, Y., Ma, X. & Guo, W. A New Characterization of Finite \( \sigma \)-Soluble \( P\sigma T \)-Groups. Sib Math J 62, 105–113 (2021). https://doi.org/10.1134/S0037446621010110
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DOI: https://doi.org/10.1134/S0037446621010110
Keywords
- finite group
- \( P\sigma T \)-group
- \( \tau_{\sigma} \)-permutable subgroup
- \( \sigma \)-soluble group
- \( \sigma \)-nilpotent group