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Rigorous Subgroup Proofs

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I need to see rigorous proofs of these propositions.

1.) Suppose [G:H] is finite. Show that there is a normal subgroup K of G
with K, a subgroup of H, such that [G:K] is finite.

2.) Suppose H is a subgroup of S_n but H is not a subgroup of A_n. Show
that [H:A_n intersect H]=2.

3.) Prove that if H,K are normal subgroups of G and HK=G, then
G/(H intersect K)is isomorphic to (G/H)X(G/K)

4.) Prove the tower law: If K is a subgroup of H a subgroup of G, then
[G:K]=[G:H][H:K]

5.) If A,B are subgroups of G and y is an element of G define
(A,B)-double coset AyB={ayb|a in A, b in B}. Show that G is the
disjoint union of its (A,B)-double cosets. Show that
|AyB|=[A^y:A^y intersect B]|B| if A,B are finite.

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Solution Summary

This provides examples of several proofs regarding subgroups in PDF form.

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