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Magazine Chapter 1: Problem 16
Show that an \(m\) -cycle is an even permutation if and only if \(m\) is odd.
Short Answer
Expert verified
An *m*-cycle is even if and only if *m* is odd.
Step by step solution
01
Understanding the Definitions
An *m*-cycle is a permutation of *m* elements where one element is moved to the place of another in a cyclic fashion. A permutation is even if it can be expressed as a product of an even number of transpositions (two-element swaps). This problem requires us to find under what conditions an *m*-cycle is even. We need to understand when an *m*-cycle results in an even number of transpositions.
02
Expressing an m-cycle as Transpositions
An *m*-cycle \( (a_1, a_2, \ldots, a_m) \) can be expressed as \( (a_1, a_2)(a_2, a_3)\ldots(a_{m-1}, a_m) \), which is comprised of \( m-1 \) transpositions. The expression of an *m*-cycle is therefore given by \[ (a_1, a_m)(a_1, a_{m-1})\ldots(a_1, a_2). \]
03
Analyzing Parity of Number of Transpositions
In an *m*-cycle, we know from the previous step that it can be expressed as \( m-1 \) transpositions. We determine the parity (even or odd nature) of the number \( m-1 \). Specifically, \( m-1 \) is even when \( m \) is odd, and \( m-1 \) is odd when \( m \) is even.
04
Conclusion with Even Permutation
Thus, the *m*-cycle permutation is even when it can be decomposed into an even count of transpositions. If \( m \) is odd, then \( m-1 \) is even, indicating that the permutation is an even permutation. Conversely, if \( m \) is even, \( m-1 \) is odd, indicating that the permutation is odd. Therefore, an *m*-cycle is an even permutation if and only if \( m \) is odd.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Even and Odd Permutations
Permutations can be categorized into two types: even and odd. This classification depends on the number of transpositions needed to express the permutation.
A permutation is considered **even** if it can be broken down into an even number of transpositions. On the other hand, it is **odd** if it consists of an odd number of these two-element swaps.
Understanding the concept of even and odd permutations is essential, especially when dealing with cycle notation in permutations. For instance:
A permutation is considered **even** if it can be broken down into an even number of transpositions. On the other hand, it is **odd** if it consists of an odd number of these two-element swaps.
Understanding the concept of even and odd permutations is essential, especially when dealing with cycle notation in permutations. For instance:
- If a permutation requires two swaps, it falls under the category of an even permutation.
- If it requires three swaps, then it is classified as an odd permutation.
Transpositions
Transpositions are the fundamental building blocks of permutations. Simply put, a transposition involves swapping two elements in a permutation while leaving the rest unchanged.
For example, consider a set of items: [1, 2, 3]. A transposition can swap 1 and 3 to form a new sequence, [3, 2, 1]. It's like a mini-shuffle between just two elements. Understanding transpositions helps in decomposing more complex permutations into simpler units. For example:
For example, consider a set of items: [1, 2, 3]. A transposition can swap 1 and 3 to form a new sequence, [3, 2, 1]. It's like a mini-shuffle between just two elements. Understanding transpositions helps in decomposing more complex permutations into simpler units. For example:
- The cycle (1, 2, 3) can be expressed as a series of transpositions: (1, 3), (1, 2).
Permutation Parity
Permutation parity pertains to whether a permutation is even or odd. It directly relates to how a permutation can be expressed as a sequence of transpositions.
For a permutation to be even, the number of transpositions must be even. Similarly, for it to be odd, the number of transpositions must be odd. Parity is a simple yet powerful concept that helps categorize and simplify permutations. Considering an *m*-cycle:
For a permutation to be even, the number of transpositions must be even. Similarly, for it to be odd, the number of transpositions must be odd. Parity is a simple yet powerful concept that helps categorize and simplify permutations. Considering an *m*-cycle:
- If in expressing this cycle, the number of transpositions (here, *m*-1) is even, the permutation itself is even. This is naturally satisfied when *m* is odd.
- If the number of transpositions is odd, then the permutation is odd, aligning with the instance where *m* is even.
