Let (Sn|n ε ω) be a sequence of finite simple non-abelian groups. A subgroup H of the complete direct product Πn Sn is said to be of small index if [ Πn Sn : H] < 2ω. We say that Πn Sn has the small index property if every subgroup H of small index is open in Πn Sn. We classify those sequences (Sn|n ε ω) of finite simple non-abelian groups such that Πn Sn has the small index property. In the course of our classification proof, we also show that if K is a field of cardinality 2ω and G is a non-trivial linear group over K, then there exists a subgroup H of G such that 1 < [G : H] ≤ ω.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory