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

From Groupprops

## Statement

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

## 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

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

## Proof

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

**To prove**: The order of divides .

**Proof**:

- Let be a -Sylow subgroup of (such a exists by fact (1)).
- is not normal in : Since divides the order of , is nontrivial. Thus, is a nontrivial normal subgroup of . Also, cannot equal because then 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.
- The subgroup is a proper subgroup of with index : The index being follows from fact (3). The
*proper*part follows from the previous step, which concluded that is not normal in . - The order of divides : This follows from the previous step and fact (4).