Let be a prime and let be a group of order . Prove that has a subgroup of order , for every with .
Hint: induction on and use the theorem: If is a prime and is a group of prime power order for some , then has a nontrivial center.
if then there's nothing to prove. so let and suppose that the claim is true for all groups of order with let G be a group of order and let we want
to show that G has a subgroup of order it's trivial for so we'll assume that we have to consider two cases:
Case 1: G is abelian. in this case choose with since G is abelian, < x > is normal in G. now G/< x > is a group of order so by induction hypothesis it has a subgroup H/<x>
of order therefore: and we're done in this case.
Case 2: G is non-abelian. let Z(G) be the center of G and then: if then by induction hypothesis Z(G) has a subgroup of order and we're done. if then
consider G/Z(G), which is a group of order by induction G/Z(G) has a subgroup K/Z(G) of order hence: