## The Nichols-Zoeller Theorem

Students of Group Theory are familiar with Lagrange's Theorem, which states
that the order of a finite group is divisible by the orders of each of its
subgroups. The most important consequence of what has become known as the
Nichols-Zoeller Theorem is the analogue for Hopf algebras: every
finite-dimensional Hopf algebra is free as a module over each of its Hopf
subalgbras. Lagrange's Theorem is a special case, since group algebras are Hopf
algebras.

The first of Kaplansky's "Ten Conjectures on Hopf Algebras", made in the
early 1970's, was that every Hopf algebra H over a field was free as a module
over any Hopf subalgebra B. Although the conjecture does not hold for H
infinite dimensional, the case in which H is finite dimensional remained a
problem of considerable interest and importance. Over the next decade,
progress on the problem was made by two of the most prominent researchers in
the area. Building on that progress, my former student Bettina Zoeller (now
Richmond) showed in her 1985 dissertation that the result was true when B was a
semisimple group algebra. Using rather more technical methods, Bettina and I,
in subsequent joint work, generalized this to the case in which B was an
arbitrary group algebra. We obtained our general result, published in 1989, by
a more sophisticated approach.

Googling the theorem, one may find:

The Nichols-Zoeller theorem is an indispensable tool in the study
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This is the content of the celebrated Nichols-Zoeller theorem ...