Let $H$ be a Hopf algebra over a field $k$ with an antipode $S$ . We will say that $K\subseteq H$ is a Hopf subalgebra if $K$ is both subalgebra and subcoalgebra of underlaying algebra and coalgebra structures of $H$ , and additionaly $S(K)\subseteq K$ . In particular a Hopf subalgebra $K\subseteq H$ is an algebra over $k$ , so $H$ may be regarded as a $K$ -module.

The Nichols-Zoeller Theorem. If $K\subseteq H$ is a Hopf subalgebra of a Hopf algebra $H$ , then $H$ is free as a $K$ -module. In particular, if $H$ is finite dimensional, then $\mathrm{dim}_{k}K$ divides $\mathrm{dim}_{k}H$ .

Remark 1. This theorem shows that Hopf algebras are very similar to groups, because this is a Hopf analogue of the Lagrange Theorem.

Remark 2. Generally this theorem does not need to hold if $H$ is only an algebra. For example, consider $H=\mathbb{M}_{n}(k)$ the matrix algebra, where $n\geq 2$ and let $T\subseteq H$ be the upper triangular matrix subalgebra. It is well known that $\mathrm{dim}_{k}H=n^2$ and $\mathrm{dim}_{k}T=\frac{n(n+1)}{2}$ . Of course $\frac{n(n+1)}{2}$ does not divide $n^2$ for $n\geq 2$ . Thus the Nichols-Zoeller Theorem does not hold for algebras.