Prove that (1 2 3 4) is not a product of 3-cycles
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Note that a 3-cycle (a b c) can be written as a product of two transpositions. For example,
(a b c) = (a b)(b c). Hence any 3-cycle is even. On the other hand, (1 2 3 4) = (1 2)(2 3)(3 4),
which shows it is odd. Any product of even permutations must be even, hence (1 2 3 4) cannot
be a product of 3-cycles
(a b c) = (a b)(b c). Hence any 3-cycle is even. On the other hand, (1 2 3 4) = (1 2)(2 3)(3 4),
which shows it is odd. Any product of even permutations must be even, hence (1 2 3 4) cannot
be a product of 3-cycles