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 · x ≠ h · 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 · x ≠ h · 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.