对称和交错群下$\mathbb{F}_2$上的部分正交展开不变性
Partial orthogonal spreads over $\mathbb{F}_2$ invariant under the symmetric and alternating groups
摘要：
设m是一个大于2的整数，V是F_2上的一个2^m维向量空间。设Q是定义在V上的具有极大Witt指数的非退化二次型，证明了对称群S_{2m+1}作为Q的等距群作用于V上，并置换大小为2m+1的部分正交扩展的成员.这意味着任何偶数阶2m或奇数阶2m+1的群在V中的部分正交展布上传递地和正则地作用。我们还证明了交替群A_9以自然的方式作用在定义在F_2上的8维向量空间上的大小为9的完全展布上。
英文摘要：
Let m be an integer greater than 2 and let V be a vector space of dimension 2^m over F_2. Let Q be a non-degenerate quadratic form of maximal Witt index defined on V. We show that the symmetric group S_{2m+1} acts on V as a group of isometries of Q and permutes the members of a partial orthogonal spread of size 2m+1. This implies that any group of even order 2m or odd order 2m+1 acts transitively and regularly on a partial orthogonal spread in V. We also show that the alternating group A_9 acts in a natural manner on a complete spread of size 9 defined on a vector space of dimension 8 over F_2.
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