Soft Computing

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

An application of subgroup lattices

Focus
  • 88 Downloads

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.

Keywords

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

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  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 http://math.berkeley.edu/~ogus/Math_113_08/supplements/cyclicity

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

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

Personalised recommendations