Why are the Integers a Cyclic Group?

Sunday, May 27th, 2012

If we follow Wikipedia in defining a cyclic group as a group in which there exists an element g in G such that G = <g> = { gn | n is an integer }, then the integers under addition are clearly a cyclic group with the generator 1. But why do we define cyclic groups that way? Or, another way of putting it, why is the definition given the name cyclic when there’s nothing cyclic about it?