Abstract
We give a lattice theoretic proof of the well-known result that a finite group G is cyclic iff G has at most one subgroup of each order dividing |G|. Consequently, we show that a division ring D is a field iff D has at most one maximal subfield.
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Notes
Note that the idea is closely related to with Frobenius conjecture on characteristic subgroup of finite group.
References
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Ogus A (2008) Math 113—Introduction to Abstract Algebra, Cyclicity of Groups, Cyclicty. Available from http://math.berkeley.edu/~ogus/Math_113_08/supplements/cyclicity
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The authors declare that they have no conflict of interest.
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Communicated by Y. Yang.
The work is partially supported by NSFC (Grant 11271040), and the Fundamental Research Funds for the Central Universities (Grant 302996).
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Chen, Y., Yang, Y. An application of subgroup lattices. Soft Comput 21, 2503–2505 (2017). https://doi.org/10.1007/s00500-017-2571-1
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DOI: https://doi.org/10.1007/s00500-017-2571-1