论文标题
基本$ p $ - 组的扩展及其在Prime指数组分类中的应用
Extension of elementary $p$-groups and its application in classification of groups of prime exponent
论文作者
论文摘要
令$ p $为素数，$ \ mathbb {z} _p = \ mathbb {z}/p \ mathbb {z} $。我们在组扩展数据及其矩阵演示的方面研究具有Abelian派生的亚组和指数$ P $的有限组。我们在以下两组之间显示一对一的对应关系：（i）2类指数$ p $和订购$ p^{m+n} $的等级以及带有派生的子组$ \ mathbb {z} _p^n $，和（ii）（ii） $ \ text {gr}（n，\ text {as} _m（\ mathbb {z} _p））/\ text {gl} _m（\ Mathbb {Z} _p） by $\text{GL}_m(\mathbb{Z}_p)$, where $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ is the set of $n$-dimensional subspaces of anti-symmetric matrices of order $m$ over $\mathbb{Z}_p$.我们给出轨道空间的描述$ \ text {gr}（2，\ text {as} _m（\ mathbb {z} _p）））/\ text {gl} _m（\ mathbb {z} _p {z} _p）$和$ m $和$ p $ p $ a ant ant anti-polent of anti-smmmetrics of poldrics of poldrics。基于此，我们显示了$ \ text {gr}（3，\ text {as} _4（\ Mathbb {z} _3））的完整代表的完整代表。 \ text {as} _4（\ Mathbb {z} _3））/\ text {gl} _4（\ Mathbb {z} _3）$和$ \ text {gr}（3，3， \ text {as} _5（\ Mathbb {z} _3））/\ text {gl} _5（\ Mathbb {z} _3）$。结果，我们获得了相应的2类指数$ p $的分类。特别是，我们恢复具有指数3的组的分类，并订购$ \ le 3^8 $。
Let $p$ be a prime number and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence between the following two sets: (i) the isoclasses of class 2 groups of exponent $p$ and order $p^{m+n}$ and with derived subgroup $\mathbb{Z}_p^n$, and (ii) the set $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))/\text{GL}_m(\mathbb{Z}_p)$ of orbits of $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ under the congruence action by $\text{GL}_m(\mathbb{Z}_p)$, where $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ is the set of $n$-dimensional subspaces of anti-symmetric matrices of order $m$ over $\mathbb{Z}_p$. We give a description of the orbit spaces $\text{Gr}(2, \text{AS}_m(\mathbb{Z}_p))/\text{GL}_m(\mathbb{Z}_p)$ for all $m$ and $p$ by applying the theory of pencils of anti-symmetric matrices. Based on this, we show complete sets of representatives of orbits of $\text{Gr}(3,\text{AS}_4(\mathbb{Z}_3))/\text{GL}_4(\mathbb{Z}_3)$, $\text{Gr}(4, \text{AS}_4(\mathbb{Z}_3))/\text{GL}_4(\mathbb{Z}_3)$ and $\text{Gr}(3, \text{AS}_5(\mathbb{Z}_3))/\text{GL}_5(\mathbb{Z}_3)$. As a consequence, we obtain a classification of corresponding class 2 groups of exponent $p$. In particular, we recover the classification of groups with exponent 3 and order $\le 3^8$.
