What about the converse? Which of the three statements do you think holds? (Don't try to prove correctness of your answer!)
For each divisor of |G|
there is an element of that order in G.
For each prime dividing
|G|
there is an element of that order in G.
Only the existence of an element
of order 1 is guaranteed.
No, there is no element of order 6 in the symmetric group on 3 letters
(a group of order 6).
True, but a proof is deferred till
a lemma in Chapter 8.
Wrong, too pessimistic! (In view of the correctness of the second statement.)