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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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.

Related facts

References

Journal references