Definition
Let be a finite set, and
be distinct elements of . The expression
denotes the cycle σ whose action is
For each index i,
where is taken to mean .
There are different expressions for the same cycle; the following all represent the same cycle:
A 1-element cycle such as (3) is the identity permutation. The identity permutation can also be written as an empty cycle, "".
Read more about this topic: Cycle Notation
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—François, Duc De La Rochefoucauld (1613–1680)
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—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)