# Transitive Sylow subgroups of finitary permutation groups

- Reference: `kourovka-21-5-transitive-sylow-subgroups-of-finitary-permutation-groups`
- Page: https://theoremdb.org/statements/kourovka-21-5-transitive-sylow-subgroups-of-finitary-permutation-groups
- Record maturity: Reviewed problem

## Problem

Let p be a prime. Let G be a transitive subgroup of the group of finitary permutations F Sym(Ω) of a set Ω, let N be a normal subgroup of G, and let S be a transitive Sylow p-subgroup of G. (a) Is it true that S ∩ N is a Sylow p-subgroup of N? (b) Is it true that SN/N is a Sylow p-subgroup of G/N? (c) Are any two transitive Sylow p-subgroups of G locally conjugate in G? Two subgroups X, Y of a group G are said to be locally conjugate if there is a locally inner automorphism φ of G such that X φ = Y. An automorphism φ of G is said to be locally inner if for every finite subset A ⊆ G there is an element g = g(A) ∈ G such that aφ = g −1 ag for all a ∈ A. This is Kourovka Notebook Problem \(21.5\).

## Status

The reviewed record remains open.

## Work

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `kourovka-21-5-transitive-sylow-subgroups-of-finitary-permutation-groups`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.