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 $\text{k}F$ by ${\text{k}}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\text{k}{C}_p$ by ${\text{k}}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\text{k}}^G$ is the Hopf algebra dual of $\k G$,$G$afiniteabelian$p$-groupand$\k$ 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 $\le p^4$.