Answer:
The permutation (1234) is not a 3-cycle
Explanation:
- To prove that (1234) is not the product of 3-cycles, we can consider the possible 3-cycles that could be used to generate (1234). If we use the 3-cycle (123), we get the permutation (1324). If we use the 3-cycle (124), we get the permutation (1432). If we use the 3-cycle (134), we get the permutation (1423). None of these permutations is equal to (1234), so (1234) cannot be the product of 3-cycles.
- To generalize this result, we can consider any permutation of n elements. If the permutation involves fewer than three elements, it cannot be the product of 3-cycles because a 3-cycle involves three elements. If the permutation involves more than three elements, it cannot be the product of 3-cycles because a 3-cycle cannot rearrange more than three elements. Therefore, any permutation of n elements that involves more or fewer than three elements cannot be the product of 3-cycles.
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