The Central Extension of an Elementary Abelian $p$-Group \\ by a Miniaml Non-Abelian $p$-Group

Abstract:

      Assume that $N$, $F$ and $G$ are groups. If there exsits $\tilde{N}$, a normal subgroup of $G$ such that $\tilde{N}\cong G$ and $G/\tilde{N}\cong F$, then $G$ is called a central extension of $N$ by $F$. In this paper, the central extension of $N$ by a minimal non-abelian $p$-group is determined, where $N$ is an elementary abelian $p$-group of order $p^3$. Together with our previous work, all central extensions of $N$ by a minimal non-abelian $p$-group is determined, where $N$ is an elementary abelian $p$-group.