One-dimensional Symmetry Group - Group Action

Group Action

Group actions of the symmetry group that can be considered in this connection are:

  • on R
  • on the set of real functions of a real variable (each representing a pattern)

This section illustrates group action concepts for these cases.

The action of G on X is called

  • transitive if for any two x, y in X there exists a g in G such that g · x = y; for neither of the two group actions this is the case for any discrete symmetry group
  • faithful (or effective) if for any two different g, h in G there exists an x in X such thatg · xh · x; for both group actions this is the case for any discrete symmetry group (because, except for the identity, symmetry groups do not contain elements that “do nothing”)
  • free if for any two different g, h in G and all x in X we have g · xh · x; this is the case if there are no reflections
  • regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any twox, y in X there exists precisely one g in G such that g · x = y.

Read more about this topic:  One-dimensional Symmetry Group

Famous quotes containing the words group and/or action:

    Caprice, independence and rebellion, which are opposed to the social order, are essential to the good health of an ethnic group. We shall measure the good health of this group by the number of its delinquents. Nothing is more immobilizing than the spirit of deference.
    Jean Dubuffet (1901–1985)

    The moment we choose to love we begin to move against domination, against oppression. The moment we choose to love we begin to move towards freedom, to act in ways that liberate ourselves and others. That action is the testimony of love as the practice of freedom.
    bell hooks (b. c. 1955)