Question

a)Express the permutation f = (1 3 2 5)(1 2 4 6)(3 6) ∈ S6 as a product of disjoint cycles. b) Let α, β ∈ S5 be given by α = (1 4 2 5 3) and β = (1 3)(2)(4 5). Find α −1β −1

Answer 1

a) The permutation f = (1 3 2 5)(1 2 4 6)(3 6) expressed as a product of disjoint cycles is f = (1 5)(2 3 6).

b) The product
`\( \alpha^(-1)\beta^(-1) \)` is given by
`\( \alpha^(-1)\beta^(-1) = (3 5)(4 2)(1) \)`.

Step-by-step explanation:

a) To express the permutation f = (1 3 2 5)(1 2 4 6)(3 6) as a product of disjoint cycles, we can consider each cycle separately and then combine them. The given permutation consists of three cycles: (1 3 2 5) , (1 2 4 6) , and (3 6) . Breaking down each cycle:

- Cycle (1 3 2 5) can be written as (1 5) because it cycles through 1, 3, 2, and 5.

- Cycle (1 2 4 6) can be written as (2 4 6) because it cycles through 1, 2, 4, and 6.

- Cycle (3 6) remains the same.

Combining these cycles, we get f = (1 5)(2 3 6) .

b) To find
`\( \alpha^(-1)\beta^(-1) \)` for
`\( \alpha = (1 4 2 5 3) \)` and
`\( \beta = (1 3)(2)(4 5) \)`, we first find the inverse of each permutation:

-
`\( \alpha^(-1) = (3 5 2 4 1) \)`

-
`\( \beta^(-1) = (5 4)(3 1)(2) \)`

Now, multiplying
`\( \alpha^(-1) \)` and
`\( \beta^(-1) \)`, we get
`\( \alpha^(-1)\beta^(-1) = (3 5)(4 2)(1) \)`.