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 <edgar-ya02408000R2210981259270001@news.math.ohio-state.edu>,
G. A. Edgar <edgar@math.ohio-state.edu> 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! *