Order has only two prime factors implies prime divisor with larger prime power is core-nontrivial except in finitely many cases
From Groupprops
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:
- and is a Fermat prime (?).
- and is a Mersenne prime (?).
- and .
Note that it may also happen that, in addition, the other prime divisor is core-nontrivial.
Related facts
References
Journal references
- A note on Burnside's other p^aq^b-theorem by Martin Coates, John Scott Rose and Michael Dwan, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Volume 14, Page 160 - 166(Year 1976): ^{Official copy}^{More info}
- On Burnside's other p^aq^b-theorem by George Glauberman, Pacific Journal of Mathematics, Volume 56, Page 469 - 476(Year 1975): This paper proves that Order has only two prime factors implies prime divisor with larger class two subgroups is core-nontrivial.^{Project Euclid page}^{More info}