The subnormal subgroup structure of the infinite symplectic group

Let R be a ring with identity and let Ω be a totally ordered set. Let Ω′ be a totally ordered set which is disjoint from and equipotent to Ω′ with ′:Ω→Ω′ an order preserving bijection. Define Ω 1 =Ω∪Ω′ and let Ω 1 , be totally ordered by inheriting the order from Ω and Ω′ and with ω<λ for all ω∈Ω and λ′∈Ω′. Let M be the free R -module R ( Ω1 ).(We define the alternate bilinear form (*, *) on M by

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