Remarks On The Regular Group Representation
The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left coset of the subgroup generated by the element.
Read more about this topic: Cayley's Theorem
Famous quotes containing the words remarks, regular and/or group:
“Where do whites fit in the New Africa? Nowhere, I’m inclined to say ... and I do believe that it is true that even the gentlest and most westernised Africans would like the emotional idea of the continent entirely without the complication of the presence of the white man for a generation or two. But nowhere, as an answer for us whites, is in the same category as remarks like What’s the use of living? in the face of the threat of atomic radiation. We are living; we are in Africa.”
—Nadine Gordimer (b. 1923)
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“I never heard of ‘Uglification,’” Alice ventured to say.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)