Examples of The Regular Group Representation
Z2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12).
Z3 = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).
Z4 = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).
S3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:
| * | e | a | b | c | d | f | permutation |
|---|---|---|---|---|---|---|---|
| e | e | a | b | c | d | f | e |
| a | a | e | d | f | b | c | (12)(35)(46) |
| b | b | f | e | d | c | a | (13)(26)(45) |
| c | c | d | f | e | a | b | (14)(25)(36) |
| d | d | c | a | b | f | e | (156)(243) |
| f | f | b | c | a | e | d | (165)(234) |
Read more about this topic: Cayley's Theorem
Famous quotes containing the words examples of, examples, regular and/or group:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“The solid and well-defined fir-tops, like sharp and regular spearheads, black against the sky, gave a peculiar, dark, and sombre look to the forest.”
—Henry David Thoreau (1817–1862)
“The virtue of dress rehearsals is that they are a free show for a select group of artists and friends of the author, and where for one unique evening the audience is almost expurgated of idiots.”
—Alfred Jarry (1873–1907)