# Order of simple non-abelian group divides factorial of index of proper subgroup

From Groupprops

## Contents

## Statement

Let be a simple non-Abelian group and be a proper subgroup of finite index in . Then, is finite and the order of divides : the factorial of the index .

## Related facts

- Poincare's theorem
- Order of simple non-Abelian group divides factorial of every Sylow number
- Order of simple non-Abelian group divides half the 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

### Stronger facts

## Related survey articles

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

## Facts used

- Simple non-Abelian group is isomorphic to subgroup of symmetric group on left coset space of proper subgroup
- Lagrange's theorem

## Proof

**Given**: A simple non-Abelian group , a proper subgroup of finite index.

**To prove**: is finite and the order of divides .

**Proof**:

- By fact (1), is isomorphic to a subgroup of .
- By fact (2), the order of divides the order of , which is .