Isomorphism Types of Hopf Algebras in a Class of Abelian Extensions.I

There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of characteristic $0$. We apply the method to calculate the number of isoclasses of commutative extensions and certain extensions of this kind of dimension $\le p^4$.