Cycle Notation - Definition

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, "".

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