Chapter 15: Groups and Symmetries
A group G is a set of objects { a, b, c, … } (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab in G (closure). The binary operation (called group multiplication) is subject to the following requirements:
There exists an element e in G called the identity element such that eg = ge = g for all g ∊ G.
For every g ∊ G there exists an inverse element g−1 in G such that gg−1 = g−1g = e.
Associative law. The identity (ab)c = a(bc) is satisfied for all a, b, c ∊ G.
If ab = ba for all a, b ∊ G we call the group commutative. If G has a finite number of elements it has finite order n(G), where n(G) is the number of elements. Otherwise, G has infinite order…