Part of the Poincare Project.
Although we've already defined them as monoids with inverses, a group is such an important concept in pure mathematics that we'll summarise here.
A group is a set G of objects with some binary operation # that maps every pair of elements of G to an element in G such that:
As we've already seen, integers under addition form a group. Integers under multiplication do not form a group because the multiplicative inverse of an integer is not an integer (e.g. inverse of 2 would be 1/2). The rationals under multiplication do not form a group either because 0 does not have an inverse. However, the non-zero rationals under multiplication do form a group.
There are many sets outside of the numbers that form groups. For example, consider the different ways you can rotate an object. Consider G to be the set of all rotations. Now consider # to be the composition of two rotations, i.e. a # b is the single rotation that is equivalent to performing rotation a after you have performed rotation b. It turns out that (G, #) forms a group.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.