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.

Read more about this topic:  Examples Of Groups

Famous quotes containing the words group, translations and/or plane:

    “The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
    —Kristin Hunter (b. 1931)

    “Woe to the world because of stumbling blocks! Occasions for stumbling are bound to come, but woe to the one by whom the stumbling block comes!”
    —Bible: New Testament, Matthew 18:7.

    Other translations use “temptations.”

    “Have you ever been up in your plane at night, alone, somewhere, 20,000 feet above the ocean?... Did you ever hear music up there?... It’s the music a man’s spirit sings to his heart, when the earth’s far away and there isn’t any more fear. It’s the high, fine, beautiful sound of an earth-bound creature who grew wings and flew up high and looked straight into the face of the future. And caught, just for an instant, the unbelievable vision of a free man in a free world.”
    —Dalton Trumbo (1905–1976)