Block Systems of finite dimensional non-cosemisimple Hopf Algebras

In this paper, we study finite dimensional non-cosemisimple Hopf algebras through its underlying coalgebra. We decompose such a Hopf algebra as a direct sum of blocks'. Blocks are closely related to each other by rules’ and form the `block system’. Through the block system, we are able to give a lower bound for $\dim H$, where $H$ is a non-cosemisimple Hopf algebras with no nontrivial skew-primitives, and a series of results about non-cosemisimple Hopf algebras of dimension $12p,15p,16p,20p\text{ and }21p$ with $p$ being a prime number.