Abstract
We prove that a finite group \(G\) is solvable if the set of its conjugacy class sizes of primary, biprimary and triprimary elements is \(\{1, m, n, mn\}\), where \(m\) and \(n\) are positive integers neither of which divides the other. Further, as a corollary, we prove that a group \(G\) is nilpotent if the set of conjugacy class sizes of its primary, biprimary and triprimary elements is \(\{1, m, n, mn\}\), where \(m\) and \(n\) are coprime integers.
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Acknowledgments
The authors are grateful to the referee for his/her outstanding efforts. The first author also wants to express his deep gratitude for the warm hospitality he received in the Departamento de Matemáticas of the Universidad Jaume I in Castellón, Spain. This research is supported by the research Project NNSF of China (Grant Nos. 11301218 and 11301219) and the Nature Science Fund of Shandong Province (No. ZR2014AM020) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336).
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Communicated by A. Constantin.
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Shao, C., Jiang, Q. A note on the solvability of finite groups with four particular conjugacy class sizes. Monatsh Math 178, 453–456 (2015). https://doi.org/10.1007/s00605-014-0728-x
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DOI: https://doi.org/10.1007/s00605-014-0728-x