TITLE: Does every group have a generator?
QUESTION [6 upvotes]: Can't ask my question specifically because it's on a take-home test, but it's critical to my current proof that I'm able to pick a generator g from a group G, and I'm not sure if I can do that.

REPLY [13 votes]: Every group has a set of generators. Groups that have one generator are called cyclic. Not every group is cyclic.