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 the, examples of, examples, regular and/or group:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“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)
“[I]n our country economy, letter writing is an hors d’oeuvre. It is no part of the regular routine of the day.”
—Thomas Jefferson (1743–1826)
“There is nothing in the world that I loathe more than group activity, that communal bath where the hairy and slippery mix in a multiplication of mediocrity.”
—Vladimir Nabokov (1899–1977)