How Do You Decompose a Random n-Cycle into 2-Cycles?

AI Thread Summary
Decomposing a random n-cycle into 2-cycles involves breaking down the cycle into pairs that represent the transitions between elements. For example, the cycle (abc) can be expressed as the product of 2-cycles (ab)(ac), indicating that 'a' transitions to 'b' and then to 'c'. This technique can be extended to longer cycles, such as (abcd), which can be represented as (ab)(ac)(ad). Understanding this decomposition helps in analyzing permutations in group theory. The method effectively simplifies the representation of complex cycles into manageable components.
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can anyone explain me the technique to decompose a random n-cycle into a bucnh of 2 cycles. Thanks in advance.
 
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Look at: (ab)(ac). Approaching this from the left, we have (1) a goes to b. (2) b goes to a in cycle one, and then a goes to c in cycle two. (3) As for c it is sent into a, cycle two.

Thus (abc)=(ab)(ac). And so forth, (abcd) = (ab)(ac)(ad), etc.
 
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