Problem 1 Let G be a simple group which has a sub group of i

Problem 1. Let G be a simple group which has a sub- group of index n. Prove that G is finite and its order divides n!

Solution

Define a group action

X:={gH;gG},G×XX,x(gH):=(xg)HX:={gH;gG},G×XX,x(gH):=(xg)H

Prove the above indeed is an action of GG on XX , and thus we get the induced homomorphism :GSym(X):GSym(X) . The kernel of this homomorphism, also known as the core of HH in GG , is characterized as the largest normal subgroup of GG contained in HH .

But since GG is simple we get then that ker=1ker=1 , and this means we can embed GG into Sym(X)Sym(X) , and this means, by Lagrange\'s Theorem, that |G|[G:H]