Multiplying Permutations in Cycle Notation

Multiply the two permutations given in cycle notation: A with three disjoint 3-cycles, and B as one 9-cycle.

When working with permutations in cycle notation, it's important to understand the underlying structures and operations. In this problem, we are tasked with multiplying two permutations: one represented by three disjoint 3-cycles and the other by a single 9-cycle. This is a quintessential exercise in understanding symmetry and the behavior of permutations.

First, recall the definition of cycle notation, which is a compact way to express permutations by identifying cycles of elements that are permuted. Disjoint cycles, such as the three 3-cycles given here, are independent of each other, meaning their composition can be computed with relative ease as they affect different parts of the set. For example, if you have cycles like (a b c), (d e f), and (g h i), each cycle acts independently on its respective set of elements. In contrast, the nature of a 9-cycle is that it involves all its elements in a single, continuous loop, which needs to be carefully traced through.

To solve the problem, follow the order of composition: typically from right to left, since functions are applied in that order. The key challenge and learning opportunity in this exercise is maintaining a clear track of where elements move as you combine these cycles. This problem is a solid practical application of understanding how to decompose permutations into simpler parts, trace their actions, and eventually reconstruct the resultant permutation from component permutations. By repeatedly practicing with varying complexities of cycle compositions, one develops a keen understanding of group operations in the context of symmetric groups.