# 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 -theorem.

## Statement

Suppose  is a group of order , where  are distinct primes with  and  are positive integers. Then,  is a Core-nontrivial prime divisor (?), except in three kinds of cases. In other words, the -Sylow-core (?) of  is nontrivial; in other words, there is a nontrivial normal -subgroup, except in the following cases:

1.  and  is a Fermat prime (?).
2.  and  is a Mersenne prime (?).
3.  and .

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