If H is a p-group show every element can be multiplied by some power to equal p.
a. No need to bring in Cauchy. For any x in G, if x is not the identity, then x has an order p^r for some r>0. What about ?
b. Further hint: prove that and , using the fact that N,L are normal in G
c. Suppose A is the subgroup of N of order 2, B is the subgroup of L of order 3. We know that everything in N and L commutes. In particular, everything in A commutes with everything in B. But A and B are cyclic. What is the subgroup AB look like? (Notice that the "subgroup" AB makes sense because and everything in A commutes with everything in B)