Order has only two prime factors implies prime divisor with larger prime power is core-nontrivial except in finitely many cases

Name
This result is sometimes termed Burnside's other $$p^aq^b$$-theorem.

Statement
Suppose $$G$$ is a group of order $$p^aq^b$$, where $$p,q$$ are distinct primes with $$p^a > q^b$$ and $$a,b$$ are positive integers. Then, $$p$$ is a fact about::core-nontrivial prime divisor, except in three kinds of cases. In other words, the $$p$$-fact about::Sylow-core of $$G$$ is nontrivial; in other words, there is a nontrivial normal $$p$$-subgroup, except in the following cases:


 * 1) $$p = 2$$ and $$q$$ is a fact about::Fermat prime.
 * 2) $$q = 2$$ and $$p$$ is a fact about::Mersenne prime.
 * 3) $$p = 2$$ and $$q = 7$$.

Note that it may also happen that, in addition, the other prime divisor is core-nontrivial.

Related facts

 * Order has only two prime factors implies solvable