Let p be a prime number. A p-group is any group whose order is a power of p. It will be shown here that if |G| =pk then G has a normal subgroup of order pm for every m between 1 and k. The proof is by induction on |G|; we therefore assume our result is true for all p-groups smaller than G. Prove parts a and b:
a. There is an element a in the center of G such that ord(a)=p.
b. is a normal subgroup of G.
c. Explain why it may be assumed that G/ has a normal subgroup of order pm-1.
d. Prove that G has a normal subgroup of order pm.