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

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


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.

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