Simple group
Tests for nonsimplicity

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 and, therefore, is not simple. If n is not a prime power, then every Sylow subgroup is proper, and, by 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.
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