Let 3 < p < q < r be odd prime numbers. We prove that the finite groups with exactly 2pqr elements of maximal order are solvable.
References
G. Y. Chen and J. Shi, “Finite groups with 30 elements of maximal order,” Appl. Categ. Structures, 16, No. 1, 239–247 (2008).
D. Gorenstein, Finite Groups, Harper & Row Press, New York (1968).
M. Herzog, “On finite simple groups of order divisible by three primes only,” J. Algebra, 120, No. 10, 383–388 (1968).
Q. Jiang and C. Shao, “Finite groups with 24 elements of maximal order,” Front. Math. China, 5, No. 4, 665–678 (2010).
V. Mazurov and E. I. Khukhro, The Kourovka Notebook: Unsolved Problems in Group Theory, 17th Edn., Russian Academy of Sciences, Siberian Division, Institute of Mathematics (2010).
G. Miller, “Addition to a theorem due to Frobenius,” Bull. Amer. Math. Soc., 11, No. 1, 6–7 (1904).
W. Shi, “Groups whose elements have given orders,” Chin. Sci. Bull., 42, Np. 21, 1761–1764 (1997).
W. Shi, “On simple K 4-group,” Chin. Sci. Bull., 36, No. 7, 1281–1283 (1991).
C. Yang, “Finite groups based on the numbers of elements of maximal order,” Chin. Ann. Math., Ser. A, 14, No. 5, 561–567 (1993) (in Chinese).
Y. Xu, J. Gao, and H. Hou, “Finite groups with 6pq elements of the largest order,” Ital. J. Pure Appl. Math., 31, 277–284 (2013).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 9, pp. 1275–1279, September, 2017.
Rights and permissions
About this article
Cite this article
Asadian, B., Ahanjideh, N. Finite Groups with 2pqr Elements of the Maximal Order. Ukr Math J 69, 1479–1484 (2018). https://doi.org/10.1007/s11253-018-1447-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-018-1447-6