Group Theory Question

I believe a cyclic group is a group where starting with a given element and applying the generator repeatedly gives all the elements of the group. So then do non-cyclic groups still have generators? And how does this work? I'm just a bit confused about what the difference is for a cyclic group - is it that the element you start with has to be the generator itself?

is it that in a non-cyclic group the generator only generates a sub group?

Non-cyclic groups can have cyclic subgroups with their own generators.

As a point of terminology, if an element x doesn't generate the whole group G, then x is not a generator (for G). But yes, in that case g will generate a cyclic subgroup of G.

Note also that even in a cyclic group, not every element x will generate the whole group. For example if you consider G ={ rotations of a square} then a 90 degree rotation generates G but a 180 degree rotation does not.

More generally, a generating set is a set of elements where taking all finite products using only elements of S generates the entire group. For example, the set of permutations of a Rubik cube has 43 quintillion elements, but is generated by the set of 6 permutations representing quarter turns of each face.

ok thank you so much this is really helpful - so just to check that my understanding is correct am i right in thinking that:
- generators only exist for cyclic groups

- for a generator the first step is completing the binary operation by applying the generator to itself and can't be just applying the generator to another element

- the existence of a generator gives rise to the fact that for a group with n (inc the identity) elements, all of them will either have period n or a factor of n