Editing Simple group (section)

==Tests for nonsimplicity==
''[[Sylow theorems#Example applications|Sylow's test]]'': Let ''n'' be a positive integer that is not prime, and let ''p'' be a prime divisor of ''n''. If 1 is the only divisor of ''n'' that is congruent to 1 modulo ''p'', then there does not exist a simple group of order ''n''.

Proof: If ''n'' is a prime-power, then a group of order ''n'' has a nontrivial [[center (group theory)|center]]<ref>See the proof in [[p-group|''p''-group]], for instance.</ref> and, therefore, is not simple. If ''n'' is not a prime power, then every Sylow subgroup is proper, and, by [[Sylow theorems|Sylow's Third Theorem]], we know that the number of Sylow ''p''-subgroups of a group of order ''n'' is equal to 1 modulo ''p'' and divides ''n''. Since 1 is the only such number, the Sylow ''p''-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

''Burnside'': A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from [[Burnside's theorem]].