An application of subgroup lattices
- 91 Downloads
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.
KeywordsSubgroup lattice Distributive lattice Cyclic group Division ring Wedderburn’s “little” Theorem
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
- Birkhoff G (1964) Lattice theory, Rev edn. AMS, Colloquium Publications, New YorkGoogle Scholar
- Burris S, Sankappanavar HP (1981) A course in universal algebra, (GTM). Springer, LondonGoogle Scholar
- 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