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

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Statement

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

Related facts

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

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

Proof

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

To prove: G is finite and the order of G divides [G:H]!.

Proof:

  1. By fact (1), G is isomorphic to a subgroup of \operatorname{Sym}(G/H).
  2. By fact (2), the order of G divides the order of \operatorname{Sym}(G/H), which is [G:H]!.