Examples of Groups - The Group of Translations of The Plane

The Group of Translations of The Plane

A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation a ∘ b as follows: first follow the prescription of b, then that of a. For instance, if

a = "move North-East for 3 miles"

and

b = "move South-East for 4 miles"

then

a ∘ b = "move East for 5 miles"

(see Pythagorean theorem for why this is so, geometrically).

The set of all translations of the plane with composition as operation forms a group:

1. If a and b are translations, then a ∘ b is also a translation.
2. Composition of translations is associative: (a ∘ b) ∘ c = a ∘ (b ∘ c).
3. The identity element for this group is the translation with prescription "move zero miles in whatever direction you like".
4. The inverse of a translation is given by walking in the opposite direction for the same distance.

This is an Abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.