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.
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.
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