Question

Suppose that H is a proper subgroup of G of index n, and that G is a simple group, that is, G has no normal subgroups except G itself and {1}. Show thatG can be embedded in Sn.

Answer 1

We are given index of H in G is n.

let X be set of all left H cosets of G.

i.e.

So, Order of

We have the group action of G on X as follows, for each

Now, We will show an homomorphism from

For each fixed , define

 such that for each x in G,



Claim : is homomorphism.

Let for any x and y in G, We have

Now, is normal subgroup of G but G is normal.

So, is trivial subgroup of G.

Hence by second isomorphism theorem, We have

This gives,

This gives can be embedded in

I hope it helps. Please feel free to revert back with further queries.