Editing Symmetric group (section)

=== Normal subgroups ===
The [[normal subgroup]]s of the finite symmetric groups are well understood. If {{math|''n'' ≤ 2}}, S<sub>''n''</sub> has at most 2 elements, and so has no nontrivial proper subgroups. The [[alternating group]] of degree ''n'' is always a normal subgroup, a proper one for {{math|''n'' ≥ 2}} and nontrivial for {{math|''n'' ≥ 3}}; for {{math|''n'' ≥ 3}} it is in fact the only nontrivial proper normal subgroup of {{math|S<sub>''n''</sub>}}, except when {{math|1=''n'' = 4}} where there is one additional such normal subgroup, which is isomorphic to the [[Klein four group]].

The symmetric group on an infinite set does not have a subgroup of index 2, as [[Giuseppe_Vitali|Vitali]] (1915<ref>{{cite journal |first=G. |last=Vitali |title=Sostituzioni sopra una infinità numerabile di elementi |journal=Bollettino Mathesis |volume=7 |pages=29–31 |date=1915 |doi= |url=}}</ref>) proved that each permutation can be written as a product of three squares. (Any squared element must belong to the hypothesized subgroup of index 2, hence so must the product of any number of squares.) However it contains the normal subgroup ''S'' of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of ''S'' that are products of an even number of transpositions form a subgroup of index 2 in ''S'', called the alternating subgroup ''A''. Since ''A'' is even a [[characteristic subgroup]] of ''S'', it is also a normal subgroup of the full symmetric group of the infinite set. The groups ''A'' and ''S'' are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by [[Luigi_Onofri|Onofri]] (1929<ref>§141, p.124 in {{cite journal |first=L. |last=Onofri |title=Teoria delle sostituzioni che operano su una infinità numerabile di elementi |journal=Annali di Matematica |volume=7 |issue=1 |pages=103–130 |date=1929 |doi=10.1007/BF02409971 |s2cid=186219904 |url=|doi-access=free }}</ref>) and independently [[J%C3%B3zef_Schreier|Schreier]]–[[Stanislaw_Ulam|Ulam]] (1934<ref>{{cite journal |last1=Schreier |first1=J. |last2=Ulam |first2=S. |title=Über die Permutationsgruppe der natürlichen Zahlenfolge |journal=Studia Math |volume=4 |issue=1 |pages=134–141 |date=1933 |doi= 10.4064/sm-4-1-134-141|url=http://matwbn.icm.edu.pl/ksiazki/sm/sm4/sm4120.pdf}}</ref>). For more details see {{harv|Scott|1987|loc=Ch. 11.3}}. That result, often called the Schreier-Ulam theorem, is superseded by a stronger one which says that the nontrivial normal subgroups of the symmetric group on a set <math>X</math> are 1) the even permutations with finite support and 2) for every cardinality <math>\aleph_0 \leq \kappa \leq |X|</math> the group of permutations with support less than <math>\kappa</math> {{harv|Dixon|Mortimer|1996|loc=Ch. 8.1}}.