What is the order of each of the following permutations? a. (124)(357) b. (124)(3567) c. (124)(3

step 1 Writing the permutation in A, we have 1 goes to 2, then 2 goes back to 1, so we've cut off that

What is the order of each of the following permutations? a....

Key Concepts

Cycle Notation

Cycle notation is a compact way to represent a permutation by denoting the cycles through which elements are permuted. In this notation, a cycle like (a b c) means that a is sent to b, b is sent to c, and c is sent back to a, while any elements not mentioned remain fixed.

Cycle Decomposition

Cycle decomposition involves expressing any permutation as a product of cycles. When working with permutations, transforming them into a collection of cycles (ideally disjoint cycles) simplifies their analysis, particularly when determining their order.

Disjoint Cycles

Disjoint cycles are cycles that have no elements in common. They are significant because disjoint cycles commute, which means that the overall behavior of a permutation composed of disjoint cycles can be analyzed by considering each individual cycle separately.

Order of a Permutation

The order of a permutation is the smallest positive integer n such that applying the permutation n times yields the identity permutation. For a permutation expressed as a product of cycles, if the cycles are disjoint, its order is determined by the individual cycles.

Least Common Multiple (LCM) in Permutations

When a permutation is decomposed into disjoint cycles, its order is the least common multiple (LCM) of the lengths of these cycles. This is because each cycle returns to its initial state after a number of applications equal to its length, so a common multiple is needed to reset all cycles simultaneously.

Handling Non-disjoint Cycles

When cycles in a permutation are not disjoint, they interact with each other, making the computation of the order more complex. In such cases, the permutation must often be rewritten as a product of disjoint cycles before the order can be determined, or alternative group theoretical techniques must be applied.

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