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
Famous quotes containing the words group and/or action:
“around our group I could hear the wilderness listen.”
—William Stafford (1914–1941)
“These people who are always briskly doing something and as busy as waltzing mice, they have little, sharp, staccato ideas.... But they have no slow, big ideas. And the fewer consoling, noble, shining, free, jovial, magnanimous ideas that come, the more nervously and desperately they rush and run from office to office and up and downstairs, thinking by action at last to make life have some warmth and meaning.”
—Brenda Ueland (1891–1985)