Groups of homeomorphisms and normal subgroups of the group of permutations
The main result is: no nontrivial proper normal subgroup of the group of permutations of a fixed set \(X\) can be the group of autohomeomorphisms of \((X,T)\) for any topology (without any additional properties) \(T\) on \(X\). For comparison, recall that any group is isomorphic to the group of all autohomeomorphisms of a topological space [\textit{J. de Groot}, Math. Ann. 138, 80--102 (1959; Zbl 0087.37802)].
scientific article; zbMATH DE number 22385