论文标题
有限$ p $ - 类$ 2 $作为中央扩展
Finite $p$-groups of class $2$ as central extensions
论文作者
论文摘要
从中央扩展的角度来处理Nilpotency 2类的有限$ P $ - 组。 给定有限的Abelian组$ g，A $，我们得出了代表$ h^2（g，a）$的元素的明确公式，计算$ h^2（g，a）$，并描述$ {\ rm end end}（g）$ {\ rm end}（g）$ and $ {\ rm end end}（\ rm end}（a）$ on $ h^2（g）$ h^2（g）$ H^2（g）这些用于提供有效的标准，以将$ g $的内态性提升到两个核心扩展之间的同态。随后，我们提出了两种应用程序来说明有用性，在情况下$ p> 2 $。 首先，我们恢复对同构$ 2 $ $ 2 $ $ p $ - 组的分类，并计算每个同构类别的自动形态组的顺序。其次，我们建立了一个非亚洲$ p $ - 订单$ p^7 $的家庭，其自动形态群体是阿贝利安。
Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extension. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and describe the actions of ${\rm End}(G)$ and ${\rm End}(A)$ on $H^2(G,A)$. These are used to provide an efficient criterion for lifting endomorphisms of $G$ to homomorphisms between two central extensions. Subsequently, we present two applications to illustrate the usefulness, in the case $p>2$. First, we recover the classification of two-generator $p$-groups of class $2$ up to isomorphism, and compute the order of the automorphism group for each isomorphism class. Second, we construct a family of non-abelian $p$-groups of order $p^7$ whose automorphism groups are abelian.
