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

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

• Order of simple non-Abelian group divides factorial of every Sylow number
• Order of simple non-Abelian group divides factorial of index of proper subgroup
• Simple non-Abelian group is isomorphic to subgroup of symmetric group on left coset space of proper subgroup
• Simple non-Abelian group is isomorphic to subgroup of alternating group on left coset space of proper subgroup of finite index

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).