The image you uploaded contains problems involving permutations and their operations, such as powers, inverses, and cycle structure. Let's break down the tasks systematically.

Here’s the explanation for some of the problems:

i. $(123)(45)(1254)^{-2}$

1. Write down the cycles:

   • $(123)$: A 3-cycle that permutes $1 \to 2 \to 3 \to 1$.
   • $(45)$: A 2-cycle that swaps $4 \leftrightarrow 5$.
   • $(1254)^{-2}$: First, find $(1254)^{-1}$, which reverses the cycle to $(4521)$. Then square it to get $(21)$ (a 2-cycle).

2. Compute the product: Combine $(123)$, $(45)$, and $(21)$ step-by-step by tracing their effects on each number.

j. $(1254)^{100}$

1. $(1254)$ is a 4-cycle, so its order is 4.
   • The powers of a cycle repeat every $n$ steps, where $n$ is the length of the cycle.
   • $100 \mod 4 = 0$, so $(1254)^{100} = (1)$, the identity permutation.

k. $|(1254)|$

1. The length of a cycle is the number of elements it permutes.
   • $(1254)$ is a 4-cycle, so $|(1254)| = 4$.

l. $|(1254)^2|$

1. First, compute $(1254)^2 = (13)(24)$ (product of disjoint 2-cycles).
   • The length of this permutation is the least common multiple of the cycle lengths.
   • $|(1254)^2| = \text{lcm}(2, 2) = 2$.

m. $(12)^{-1}$

1. The inverse of a 2-cycle is the cycle itself.
   • $(12)^{-1} = (12)$.

Would you like detailed solutions for the remaining questions or assistance with a specific part?

Related Questions:

1. How do you compute the inverse of a cycle notation permutation?
2. What is the general process to find powers of a permutation?
3. How do you determine the order of a permutation?
4. Why is the product of disjoint cycles independent of order?
5. What role does modular arithmetic play in simplifying powers of permutations?

Tip:

When working with cycle notation, always track how elements move under each cycle carefully, and simplify step-by-step.