Question
23
123
123
Given these permutations 푏 = (1
123) , 푐 = (
312) and 푑 = (
231).
a) Write these permutations in cycle notations.
b) Find the inverse of each permutation.
c) Find the order of each permutation.
d) Determine for each permutation whether it is even permutation or odd permutation.
123
123
Given these permutations 푏 = (1
123) , 푐 = (
312) and 푑 = (
231).
a) Write these permutations in cycle notations.
b) Find the inverse of each permutation.
c) Find the order of each permutation.
d) Determine for each permutation whether it is even permutation or odd permutation.
Answers
GPT-4o mini
To solve this problem, we need to interpret the permutations given in two-line notation and then follow the steps outlined in the question.
### Given Permutations
1. \( b = (1\, 123) \)
2. \( c = (312) \)
3. \( d = (231) \)
### a) Write these permutations in cycle notation.
**Permutation \( b = (1\, 123) \)**:
- In expanded form, it means \(1\) maps to \(1\), \(2\) maps to \(1\), \(3\) maps to \(2\), and \(1\) maps to \(3\).
- This permutation can be expressed as:
\[
(1)(2\, 3)
\]
However, we usually do not write the single-element cycles, so the cycle notation becomes \((2\, 3)\).
**Permutation \( c = (312) \)**:
- This means \(3 \to 1\), \(1 \to 2\), and \(2 \to 3\).
- This can be expressed in cycle notation as:
\[
(1\, 2\, 3)
\]
**Permutation \( d = (231) \)**:
- This means \(2 \to 3\), \(3 \to 1\), and \(1 \to 2\).
- The cycle notation is:
\[
(1\, 2\, 3)^{-1} \text{ (which is }(3\, 2\, 1)\text{)} = (1\, 3\, 2)
\]
So, the cycle notations are:
- \( b = (2\, 3) \)
- \( c = (1\, 2\, 3) \)
- \( d = (1\, 3\, 2) \)
### b) Find the inverse of each permutation.
1. **Inverse of \( b = (2\, 3) \)**:
- The inverse of a 2-cycle is the cycle itself:
\[
b^{-1} = (2\, 3)
\]
2. **Inverse of \( c = (1\, 2\, 3) \)**:
- The inverse of a cycle can be found by reversing the order:
\[
c^{-1} = (3\, 2\, 1)
\]
3. **Inverse of \( d = (1\, 3\, 2) \)**:
- Similarly, reverse the order for the inverse:
\[
d^{-1} = (2\, 3\, 1)
\]
### c) Find the order of each permutation.
- The order of a permutation is the least common multiple (LCM) of the lengths of its cycles.
1. **Order of \( b = (2\, 3) \)**:
- Length of cycle is \(2\).
\[
\text{Order of } b = 2
\]
2. **Order of \( c = (1\, 2\, 3) \)**:
- Length of cycle is \(3\).
\[
\text{Order of } c = 3
\]
3. **Order of \( d = (1\, 3\, 2) \)**:
- Length of cycle is \(3\).
\[
\text{Order of } d = 3
\]
### d) Determine for each permutation whether it is even permutation or odd permutation.
- An even permutation is one that can be expressed as the product of an even number of transpositions (2-cycles). An odd permutation is one that can be expressed as the product of an odd number of transpositions.
1. **Permutation \( b = (2\, 3) \)**:
- Being a single transposition, it is an **odd permutation**.
2. **Permutation \( c = (1\, 2\, 3) \)**:
- Can be expressed as \((1\, 2)(1\, 3)\) (two transpositions):
- Hence, it is an **even permutation**.
3. **Permutation \( d = (1\, 3\, 2) \)**:
- Can be expressed as \((2\, 3)(1\, 3)\) (two transpositions):
- Hence, it is also an **even permutation**.
### Summary
- a)
- \( b = (2\, 3) \)
- \( c = (1\, 2\, 3) \)
- \( d = (1\, 3\, 2) \)
- b)
- \( b^{-1} = (2\, 3) \)
- \( c^{-1} = (3\, 2\, 1) \)
- \( d^{-1} = (2\, 3\, 1) \)
- c)
- Order of \( b = 2 \)
- Order of \( c = 3 \)
- Order of \( d = 3 \)
- d)
- \( b \) is odd.
- \( c \) is even.
- \( d \) is even.
### Given Permutations
1. \( b = (1\, 123) \)
2. \( c = (312) \)
3. \( d = (231) \)
### a) Write these permutations in cycle notation.
**Permutation \( b = (1\, 123) \)**:
- In expanded form, it means \(1\) maps to \(1\), \(2\) maps to \(1\), \(3\) maps to \(2\), and \(1\) maps to \(3\).
- This permutation can be expressed as:
\[
(1)(2\, 3)
\]
However, we usually do not write the single-element cycles, so the cycle notation becomes \((2\, 3)\).
**Permutation \( c = (312) \)**:
- This means \(3 \to 1\), \(1 \to 2\), and \(2 \to 3\).
- This can be expressed in cycle notation as:
\[
(1\, 2\, 3)
\]
**Permutation \( d = (231) \)**:
- This means \(2 \to 3\), \(3 \to 1\), and \(1 \to 2\).
- The cycle notation is:
\[
(1\, 2\, 3)^{-1} \text{ (which is }(3\, 2\, 1)\text{)} = (1\, 3\, 2)
\]
So, the cycle notations are:
- \( b = (2\, 3) \)
- \( c = (1\, 2\, 3) \)
- \( d = (1\, 3\, 2) \)
### b) Find the inverse of each permutation.
1. **Inverse of \( b = (2\, 3) \)**:
- The inverse of a 2-cycle is the cycle itself:
\[
b^{-1} = (2\, 3)
\]
2. **Inverse of \( c = (1\, 2\, 3) \)**:
- The inverse of a cycle can be found by reversing the order:
\[
c^{-1} = (3\, 2\, 1)
\]
3. **Inverse of \( d = (1\, 3\, 2) \)**:
- Similarly, reverse the order for the inverse:
\[
d^{-1} = (2\, 3\, 1)
\]
### c) Find the order of each permutation.
- The order of a permutation is the least common multiple (LCM) of the lengths of its cycles.
1. **Order of \( b = (2\, 3) \)**:
- Length of cycle is \(2\).
\[
\text{Order of } b = 2
\]
2. **Order of \( c = (1\, 2\, 3) \)**:
- Length of cycle is \(3\).
\[
\text{Order of } c = 3
\]
3. **Order of \( d = (1\, 3\, 2) \)**:
- Length of cycle is \(3\).
\[
\text{Order of } d = 3
\]
### d) Determine for each permutation whether it is even permutation or odd permutation.
- An even permutation is one that can be expressed as the product of an even number of transpositions (2-cycles). An odd permutation is one that can be expressed as the product of an odd number of transpositions.
1. **Permutation \( b = (2\, 3) \)**:
- Being a single transposition, it is an **odd permutation**.
2. **Permutation \( c = (1\, 2\, 3) \)**:
- Can be expressed as \((1\, 2)(1\, 3)\) (two transpositions):
- Hence, it is an **even permutation**.
3. **Permutation \( d = (1\, 3\, 2) \)**:
- Can be expressed as \((2\, 3)(1\, 3)\) (two transpositions):
- Hence, it is also an **even permutation**.
### Summary
- a)
- \( b = (2\, 3) \)
- \( c = (1\, 2\, 3) \)
- \( d = (1\, 3\, 2) \)
- b)
- \( b^{-1} = (2\, 3) \)
- \( c^{-1} = (3\, 2\, 1) \)
- \( d^{-1} = (2\, 3\, 1) \)
- c)
- Order of \( b = 2 \)
- Order of \( c = 3 \)
- Order of \( d = 3 \)
- d)
- \( b \) is odd.
- \( c \) is even.
- \( d \) is even.