DSER初等正交群的正规性
Normality of DSER elementary orthogonal group
摘要：
设$(Q, q)$是交换环$R$上的二次空间，其中$2$可逆，考虑正交群群$O_R(Q \perp H(R)^m)$的Dickson-Siegel-Eichler-Roy子群$EO_{R}(Q, H(R)^{m})$，秩为$Q= n \geq 1$和$m\geq 2$ .证明了对于所有$m\geq 2$，$EO_{R}(Q, H(R)^{m})$是$O_R(Q \perp H(R)^m)$的正规子群.我们还证明了DSER群$EO_{R}(Q, H(P))$是$O_{R}(Q \perp H(P))$的正规子群，其中$Q$和$H(P)$是交换环$R$上的二次空间，秩分别为$(Q) \ge 1$和$(P) \ge 2$ .
英文摘要：
Let $(Q, q)$ be a quadratic space over a commutative ring $R$ in which $2$ is invertible, and consider the Dickson--Siegel--Eichler--Roy's subgroup $EO_{R}(Q, H(R)^{m})$ of the orthogonal group $O_R(Q \perp H(R)^m)$, with rank $Q= n \geq 1$ and $m\geq 2$. We show that $EO_{R}(Q, H(R)^{m})$ is a normal subgroup of $O_R(Q \perp H(R)^m)$, for all $m\geq 2$. We also prove that the DSER group $EO_{R}(Q, H(P))$ is a normal subgroup of $O_{R}(Q \perp H(P))$, where $Q$ and $H(P)$ are quadratic spaces over a commutative ring $R$, with rank $(Q) \ge 1$ and rank $(P) \ge 2$.
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