Finite $p$-groups of class two with a large multiple holomorph
Finite $p$-groups of class two with a large multiple holomorph
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具有大型多重全息图的第二类有限 $p$ 群
DOI：
10.1016/j.jalgebra.2022.11.013
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发表时间：
2023
影响因子：
0.9
通讯作者：
Tsang Cindy (Sin Yi)
中科院分区：
文献类型：
--
作者：
Caranti A.;Tsang Cindy (Sin Yi)
设G为任意群。本文研究了群G的各种族的多重全形被G的全形的商群T（G）。本文设G是一个对任意奇素数p为第二类的有限p-群，在这种情况下，T（G）可以用某些双线性形式来研究。对任意n≥ 4，给出了pn+（n2）阶G的例子，使得T（G）包含一个同构于GLn（Fp）× GL（n2）− n（Fp）的子群.对于有限p-群G，迄今已知的T（G）阶素因子都来自p（p− 1）。我们的例子表明，T（G）的阶也可以有其他素因子。实际上，我们可以把任意有限群嵌入到T（G）中，以适当地选择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≥ 4, we exhibit examples of G of order p n+(n 2) such that T (G) contains a subgroup isomorphic to GL n (F p)× GL (n 2)− n (F p). For finite p-groups G, the prime factors of the order of T (G) which were 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.