Soft Computing

, Volume 21, Issue 10, pp 2503–2505 | Cite as

An application of subgroup lattices



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.


Subgroup lattice Distributive lattice Cyclic group Division ring Wedderburn’s “little” Theorem 


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Conflict of interest

The authors declare that they have no conflict of interest.


  1. Birkhoff G (1964) Lattice theory, Rev edn. AMS, Colloquium Publications, New YorkGoogle Scholar
  2. Burris S, Sankappanavar HP (1981) A course in universal algebra, (GTM). Springer, LondonGoogle Scholar
  3. Lam TY (1991) A first course in noncommutative rings. Springer, LondonCrossRefMATHGoogle Scholar
  4. Ogus A (2008) Math 113—Introduction to Abstract Algebra, Cyclicity of Groups, Cyclicty. Available from

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of FoundationFujian Commercial CollegeFuzhouChina
  2. 2.Department of MathematicsBeihang UniversityBeijingChina

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