Question

Express the following as a product of disjoint cycle. Check them for being even or odd permutation and find the inverse of each of them in S7
1. (1 4 7) (2 6 5) (2 4 1) (5 6 7)
2. (7 1) (2 6 5) (2 4 1) (7 5 6)

Answer 1

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 $(24)(27)(16)=(274)(16).$