Let us express the permutations as a product of disjoint cycle using a rule for the product of cycles (from right to left): "1\\to1\\to2\\to6\\to 6, 6\\to7\\to7\\to 7\\to1" and so on. Therefore,
"(1 4 7) (2 6 5) (2 4 1) (5 6 7)=(16)(27)" is a product of disjoint cycles. This permutation consist of two transpositions, and thus is even. Since a transposition has order 2, the inverse of "(16)(27)" is "(27)(16)."
By analogy, "(7 1) (2 6 5) (2 4 1) (7 5 6)=(16)(247)" is a product of disjoint cycles. Since "(16)(247)=(16)(27)(24)", this permutation is odd. The inverse of "(7 1) (2 6 5) (2 4 1) (7 5 6)=(16)(27)(24)" is "(24)(27)(16)=(274)(16)."
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Dear arci azarcon, please use the panel for submitting a new question.
2. Prove that a cycle of length l is even if l is odd.
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