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