On Certain Closed Normal Subgroups of Free Profinite Groups of Countably Infinite Rank

Abstract

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R _Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R _Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R _Δ(F) is solvable.

Key Words:

• Profinite groups
• Embedding problems

Mathematics Subject Classification:

• 20E18

Acknowledgments

The author would like to express her sincere gratitude to Professor Yuichiro Taguchi who gave her invaluable assistance. Also the author would like to express her special thanks to Professor Akio Tamagawa whose repeated advice was most enlightening. In addition, the author is grateful to Professor Keiichi Komatsu for his suggestion on the proof of Theorem 0.2.

Notes

^#Communicated by E. Zelmanov.