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:
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)
“The Oriental philosophy approaches easily loftier themes than the modern aspires to; and no wonder if it sometimes prattle about them. It only assigns their due rank respectively to Action and Contemplation, or rather does full justice to the latter. Western philosophers have not conceived of the significance of Contemplation in their sense.”
—Henry David Thoreau (1817–1862)