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:

- the operation is
*associative*: (a # b) # c = a # (b # c) for all a, b, c in G - there is an
*identity*element e in G where x # e = e # x = x for all x in G - every element a has an
*inverse*b such that a # b = b # a = e (where e is the identity element)

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

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.