From: greg@math.ucdavis.edu (Greg Kuperberg) Newsgroups: sci.math.research Subject: Re: linear algebra Date: 23 Oct 1998 15:03:58 -0700 In article , G. A. Edgar wrote: >(1) Let A be an algebra over a field F of dimension n. If B > is a subalgebra, must the dimension of B be a divisor of n? Of course the answer is no, but there are non-trivial variants of the question for which the answer is yes. My favorite is the "Hopf Algebra Lagrange's Theorem" of Nichols and Richmond (nee Zoeller). The theorem says that if A is a finite-dimensional Hopf algebra and B is a Hopf subalgebra, A is a free B-module. This implies divisibility of dimensions. It is interesting that the same result holds for nested division algebras as well as for Hopf algebras. It never occurred to me to compare these two facts. Is there possibly a mutual generalization? -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the xxx Math Archive Front at http://front.math.ucdavis.edu/ \/ * 6407+901 e-prints and counting! *