初等交换$p$群被内交换$p$群的中心扩张

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

中文关键词:  中心扩张  内交换$p$群  合同

英文关键词:central extension  minimal non-abelian $p$-groups  congruent

中文摘要:

      设$N$, $F$和$G$是群.若存在$G$的正规子群$\tilde{N}$使得$\tilde{N}\cong G$和$G/\tilde{N}\cong F$成立, 则称$G$为$N$被$F$的中心扩张. 本文决定了$p^3$阶初等交换$p$群被内交换$p$群的中心扩张. 与其它工作一起, 初等交换$p$群被内交换$p$群的中心扩张已经全部被决定.

英文摘要:

      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.