Group theory problem exceedinly difficult and no one can solve it. can you?

[betty2301]

Homework Statement

~p.s. it should be H/N is abelian, not H being abelian.

Homework Equations

subgroup

The Attempt at a Solution

for a) i have some idea
for b) i have no idea.

help~
:)

 

[jbunniii]

Regarding (b), any subgroup of a solvable group must be solvable. (Why?) What interesting property do you know about a certain subgroup of $S_n$ for $n \geq 5$?

 

[jbunniii]

Regarding (a), what happens to the cycle structure of an element of $S_n$ if you conjugate it by another element?