Group Orders That Imply a Nontrivial p-Core
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Given a prime number p and a natural number m not divisible by p, we propose the problem of finding the smallest number r0 such that for r ≥ r0, every group G of order p rm has a non-trivial normal p-subgroup. We prove that we can explicitly calculate the number r0 in the case where every group of order p rm is solvable for all r, and we obtain the value of r0 for a case where m is a product of two primes.
Abstract
Given a prime number p and a natural number m not divisible by p, we propose the problem of finding the smallest number r0 such that for r ≥ r0, every group G of order p rm has a non-trivial normal p-subgroup. We prove that we can explicitly calculate the number r0 in the case where every group of order p rm is solvable for all r, and we obtain the value of r0 for a case where m is a product of two primes.
참고문헌 (5)
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- D. Gorenstein, "Finite groups" (1980), Chelsea Publishing Co
- D. Quillen, "Homotopy properties of the poset of nontrivial p-subgroups of a group", Adv. in Math, 28(2), pp.101~128 (1987)
- J. S. Rose, "A course on group theory" (1994), Dover Publications Inc
- P. J. Webb, "Subgroup complexes", Proc. Sympos. Pure Math, 47(1) (1987)