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:
“If the Russians have gone too far in subjecting the child and his peer group to conformity to a single set of values imposed by the adult society, perhaps we have reached the point of diminishing returns in allowing excessive autonomy and in failing to utilize the constructive potential of the peer group in developing social responsibility and consideration for others.”
—Urie Bronfenbrenner (b. 1917)
“Rash actions are seldom committed in isolation. With the first rash action we always do too much. So we usually go on to commit a second one—and then we do too little.”
—Friedrich Nietzsche (1844–1900)