How would you go about showing a group is cyclic in group theory?

Answer 1

If the number of members of a group is prime, you know for sure the group is cyclic. Otherwise, show that theres is a member g such that for each member h there is an integer n such that g^n=h.

Answer 2

Amit is right - usually one finds an explicit generator of the group.

Sometimes, that's not possible. E.g. to prove that the multiplicative group of a fiinite field is cyclic, one pulls out this result:

Theorem: if G is an abelian group, such that for each n>0, the number of solutions to x^n= identity is at most n, then G is cyclic.