This section is fascinating. It establishes some of the basic properties of groups, but the one that I find particularly interesting is in Theorem 7.8 on the order of elements. Each element may be the identity element if acted upon by itself enough times ( if for some positive integer k, then the element a has finite order) or this will never happen (if for every positive integer k, then the element a is of infinite order). There are a few examples (and two theorems) of when it is possible to tell if an integer a is of finite or infinite order, but are there more generalized cases of this? How can this be applied? One idea that comes to mind is data transfer and cryptography, but how would it be applied?