An application of subgroup lattices
- 84 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
- 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
© Springer-Verlag Berlin Heidelberg 2017