Editing Sylow theorems (section)

=== Consequences ===
The Sylow theorems imply that for a prime number <math>p</math> every Sylow <math>p</math>-subgroup is of the same order, <math>p^n</math>. Conversely, if a subgroup has order <math>p^n</math>, then it is a Sylow <math>p</math>-subgroup, and so is conjugate to every other Sylow <math>p</math>-subgroup. Due to the maximality condition, if <math>H</math> is any <math>p</math>-subgroup of <math>G</math>, then <math>H</math> is a subgroup of a <math>p</math>-subgroup of order <math>p^n</math>.

An important consequence of Theorem 2 is that the condition <math>n_p = 1</math> is equivalent to the condition that the Sylow <math>p</math>-subgroup of <math>G</math> is a [[normal subgroup]] (Theorem 3 can often show <math>n_p = 1</math>). However, there are groups that have proper, non-trivial normal subgroups but no normal Sylow subgroups, such as <math>S_4</math>. Groups that are of prime-power order have no proper Sylow <math>p</math>-subgroups.

The third bullet point of the third theorem has as an immediate consequence that <math>n_p</math> [[divides]] <math>|G|</math>.