Let $X$ be an infinite set and let $\operatorname{Sym}(X)$ be the symmetric group of $X.$ Let $N$ denote the set of all permutations $\pi\in\operatorname{Sym}(X),$ such that the complement of the set of fixed points of $\pi$ has cardinality strictly less than $\operatorname{card}(X).$
A theorem of Baer is cited in a paper I'm trying to read, which says or implies that $N\lhd \operatorname{Sym}(X)$ and is maximal in the family of normal subgroups of $\operatorname{Sym}(X).$
Baer's paper is in German so I cannot read it. I don't have much trouble with the fact that $N\lhd \operatorname{Sym}(X),$ but the maximality eludes me. Could you please help me with it?
If anyone's interested, the cited paper is
Baer, R., Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich, Studia Math. 5 (1934), 15–17.