Let be an element of a finite group . What is meant by the order of ? Prove that the order of must divide the order of . [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]
If is a group of order , and is a divisor of with , is it always true that must contain an element of order ? Justify your answer.
Prove that if and are coprime then the group is cyclic.
If and are not coprime, can it happen that is cyclic?
[Here denotes the cyclic group of order .]