Abstract
Let \(\sigma\) be a partition of the set of all primes \(\mathbb{P}\). Let G be a finite group and \(\mathfrak{F}\) be a Fitting class of finite groups. In the theory of formations of finite soluble groups, a well known result of Bryce and Cossey is: a local formation \(\mathfrak{F}\) is a Fitting class if and only if every value of the canonical formation function F of \(\mathfrak{F}\) is a Fitting class. In this paper, we give the dual theory of the result of Bryce and Cossey. We proved that an \(\sigma\)-local Fitting class \(\mathfrak{F}\) is a formation if and only if every value of the canonical \(\sigma\)-local \(H_{\sigma}\)-function of \(\mathfrak{F}\) is a formation.
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The authors thank the referees for their careful reading and helpful comments.
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Chi Zhang is supported by Fundamental Research Funds for the Central Universities (No. 2020QN20), NSFC of China (No. 12001526) and Natural Science Foundation of Jiangsu Province, China (No. BK20200626).
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Hussain, M.T., Zafar, Z.U.A. & Zhang, C. On dualization of a result of Bryce and Cossey theory. Acta Math. Hungar. 165, 40–47 (2021). https://doi.org/10.1007/s10474-021-01168-0
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DOI: https://doi.org/10.1007/s10474-021-01168-0
Key words and phrases
- finite group
- Fitting class
- Lockett class
- \(\sigma\)-local Fitting class
- Hartley \(\sigma\)-function
- Lockett formation