Editing Symmetric group (section)

=== Maximal subgroups ===
{{expand section|date=September 2009}}

The [[maximal subgroup]]s of {{math|S<sub>''n''</sub>}} fall into three classes: the intransitive, the imprimitive, and the primitive.  The intransitive maximal subgroups are exactly those of the form {{math|S<sub>''k''</sub> × S<sub>''n''–''k''</sub>}} for {{math|1 ≤ ''k'' < ''n''/2}}.  The imprimitive maximal subgroups are exactly those of the form {{math|S<sub>''k''</sub> wr S<sub>''n''/''k''</sub>}}, where {{math|2 ≤ ''k'' ≤ ''n''/2}} is a proper divisor of ''n'' and "wr" denotes the [[wreath product]].  The primitive maximal subgroups are more difficult to identify, but with the assistance of the [[O'Nan–Scott theorem]] and the [[classification of finite simple groups]], {{harv|Liebeck|Praeger|Saxl|1988}} gave a fairly satisfactory description of the maximal subgroups of this type<!-- though beware of typos in the low degrees-->, according to {{harv|Dixon|Mortimer|1996|p=268}}.