I know the order of a group is the size of the group, ie the number of elements. But what does it mean for an element of that group to have order?
Also, what are the precise definitions for
1) "element of a finite order of a group" and 2) order of an element of a group (assuming that the element has finite order)
If I remember correctly, the order of $\mathbb{Z}$ is one, however the order of the elements in this group have order infinity. Why is that? (Also I dont think $\mathbb{Z}$ is a group in the first place, is it? )
I would also like to ask another question if thats okay:
What is a cyclic group (and a precise definition for it as well) ?