Final answer:
To transform between permutation notations, apply the rules of cycle notation for permutation, use transposition notation to detail the swaps, list images in order for permutation notation, and use exponents in exponential notation to indicate repeated cycles. The order in cycle and transposition notation is unique and independent for disjoint cycles.
Step-by-step explanation:
The student's question involves several forms of expressing a permutation, so we need a specific example to fully answer it. However, I can provide a general explanation on how to convert between the different notations of permutations.
- Cycle notation represents a permutation by cycles indicating which elements are permuted. For example, the cycle (1 2 3) means 1 goes to 2, 2 goes to 3, and 3 goes back to 1.
- Transposition notation breaks down a permutation into a product of two-element swaps, or transpositions, such as (1 2)(2 3) which is equivalent to the cycle (1 2 3).
- Permutation notation list the images of the elements in order, such as [2 3 1] for the cycle (1 2 3).
- Exponential notation may refer to writing cycles with exponents to indicate the number of times a cycle is applied, such as (1 2 3)2, though this usage is less common.
To compose cycles, simply apply each cycle in order to the elements, and for disjoint cycles, they can be multiplied in any order since they affect separate elements. It's also worth noting that every permutation can be written as a product of disjoint cycles or transpositions, and these are unique up to the order of the cycles or transpositions.