论文标题
有限的$ P $ - 级别的二等小组，具有较大的多个霍洛姆
Finite $p$-groups of class two with a large multiple holomorph
论文作者
论文摘要
让$ g $是任何小组。已针对各组$ g $的家族调查了$ g $ holomorph的多个Holomorph的商组$ t（g）$。在本文中，对于任何奇数$ p $，我们将采用$ g $作为有限的$ p $ - 第二类，在这种情况下，可以使用某些双线性表格来研究$ t（g）$。对于任何$ n \ geq 4 $，我们展示了$ g $ of订单的示例$ p^{n+{n+oplece 2}} $，使得$ t（g）$包含一个子组isomorphic to \ begin \ begin {equination*} \ pertaratOrname {epperatorname {gl} _n（gl} _n（gl} _n（\ mathbb {\ mathbb {\ mathbb {f} _p） \ operatoTorName {gl} _ {\ binom {n} {2} -n}（\ Mathbb {f} _p）。\ end end {equation {equation*}，有限$ p $ -ggroups $ -groups $ g $，$ t（g）的阶段的主要因素是$ t（g）的典型因素，$ t（g）$ t（g）$ t（g）$ t（g）$ p $ p（p（p））$ p（p（p p）。我们的示例表明，$ t（g）$的顺序也可以具有其他主要因素。实际上，我们可以将任何有限的组嵌入$ t（g）$中，以选择$ g $。
Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd prime $p$, in which case $T(G)$ may be studied using certain bilinear forms. For any $n\geq 4$, we exhibit examples of $G$ of order $p^{n+{n\choose 2}}$ such that $T(G)$ contains a subgroup isomorphic to \begin{equation*} \operatorname{GL}_n(\mathbb{F}_p) \times \operatorname{GL}_{\binom{n}{2}-n}(\mathbb{F}_p).\end{equation*} For finite $p$-groups $G$, the prime factors of the order of $T(G)$ which are known so far all came from $p(p-1)$. Our examples show that the order of $T(G)$ can have other prime factors as well. In fact, we can embed any finite group into $T(G)$ for a suitable choice of $G$.
