# 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 Core-nontrivial prime divisor (?), except in three kinds of cases. In other words, the $p$-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 Fermat prime (?).
2. $q = 2$ and $p$ is a Mersenne prime (?).
3. $p = 2$ and $q = 7$.

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