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Subgroup proofs

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Let G be a group, not necessarily finite, and let H be subgroup G.
(a) Prove that U = intersection of all x in G xHx^-1 is the largest
normal subgroup of G contained in H.
(b) Show that no proper subgroup H of A_5 contains six distinct Sylow
5-subgroups.

I need a detailed rigorous proof of this to study please.

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Solution Summary

This provides an example of completing a proof about the largest normal subgroup in a group and Sylow 5-subgroups.

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(a)
If N is a normal subgroup of G contained in H, then the intersection of all the conjugates of N in G is contained in the intersection of all the ...

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