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.
Read more about this topic: One-dimensional Symmetry Group
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