Proof the isomorphism between symmetric group & subgroup ?

Question

For the symmetric group S(2k) there are two equal subsets A = {1,....,k} and A' = {k+1,.....,2k}. Let L be the subgroup of all permutations r of S(2k) with r(A) = A or r(A)= A' and r(A') = A or r(A') = A'. Let M ≤ L for all r' of S(2k) with r'(A) = A and r'(A') = A'.

Proof:

L/M is isomorph to S(2k)

What I know: L/M := {lM :l of L} but I don't know how to show the statement. What's easier: to show that it is injective and surjective or the inverse function, and if the second one is easier how do I see the inverse function ?

Thanks for advice.sub

Answer 1

By its very definition, $L$ acts (in fact, nontrivially) on the set $\{A,A'\}$ and $M$ is the kernel of this action.