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Contents 1 Name 2 Statement 3 Related facts 4 References 4.1 Journal references

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 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 the other prime divisor is core-nontrivial.

Related facts

• Order has only two prime factors implies solvable

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 copyMore 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 pageMore info}