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…