Order of simple non-abelian group divides half the factorial of every Sylow number

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Statement

Suppose G is a finite simple non-Abelian group and p is a prime dividing the order of G. Let n_p denote the p-Sylow number of G: the number of p-Sylow subgroups of G. Then, the order of G divides n_p!/2.

Related facts

Related survey articles

Small-index subgroup technique: The use of this and similar results to prove that groups satisfying certain conditions (e.g., conditions on the order) cannot be simple.

Facts used

  1. Sylow subgroups exist
  2. Prime power order implies not centerless
  3. Sylow number equals index of Sylow normalizer
  4. Order of simple non-Abelian group divides half the factorial of index of proper subgroup

Proof

Given: A finite simple non-Abelian group G, a prime p dividing the order of G. n_p is the number of p-Sylow subgroups.

To prove: The order of G divides n_p!/2.

Proof:

  1. Let P be a p-Sylow subgroup of G (such a P exists by fact (1)).
  2. P is not normal in G: Since p divides the order of G, P is nontrivial. Thus, P is a nontrivial normal subgroup of G. Also, P cannot equal G because then G would be a group of prime power order, hence have a nontrivial center by fact (2), contradicting the assumption that it is simple non-Abelian.
  3. The subgroup N_G(P) is a proper subgroup of G with index n_p: The index being n_p follows from fact (3). The proper part follows from the previous step, which concluded that P is not normal in G.
  4. The order of G divides n_p!/2: This follows from the previous step and fact (4).