Question
Given these permutations 𝑎 = (
1 2 3
1 2 3
) , 𝑏 = (
1 2 3
3 1 2
) and 𝑐 = (
1 2 3
2 3 1
).
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.
1 2 3
1 2 3
) , 𝑏 = (
1 2 3
3 1 2
) and 𝑐 = (
1 2 3
2 3 1
).
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 analyze the given permutations \(a\), \(b\), and \(c\), let's first rewrite them in the standard notation where a permutation \(\sigma\) maps each element of its domain to another element in its codomain.
### a) Write these permutations in cycle notation.
Given permutations are:
1. **Permutation \(a\)**:
\[
a = \begin{pmatrix}
1 & 2 & 3 \\
1 & 2 & 3
\end{pmatrix}
\]
This means \(1 \to 1\), \(2 \to 2\), and \(3 \to 3\). In cycle notation, it is the identity permutation.
\[
a = (1)(2)(3) \quad \text{or simply } a = e
\]
2. **Permutation \(b\)**:
\[
b = \begin{pmatrix}
1 & 2 & 3 \\
3 & 1 & 2
\end{pmatrix}
\]
This means \(1 \to 3\), \(2 \to 1\), and \(3 \to 2\). In cycle notation, it can be written as:
\[
b = (1 \, 3 \, 2)
\]
3. **Permutation \(c\)**:
\[
c = \begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1
\end{pmatrix}
\]
This means \(1 \to 2\), \(2 \to 3\), and \(3 \to 1\). In cycle notation, it is:
\[
c = (1 \, 2 \, 3)
\]
### b) Find the inverse of each permutation.
1. **Inverse of \(a\)**:
The identity permutation is self-inverse:
\[
a^{-1} = a = e
\]
2. **Inverse of \(b\)**:
To find the inverse, we reverse the mappings:
- From \(2 \to 1\), we have \(1 \leftarrow 2\).
- From \(1 \to 3\), we have \(3 \leftarrow 1\).
- From \(3 \to 2\), we have \(2 \leftarrow 3\).
Thus, the inverse \(b^{-1}\) can be expressed as:
\[
b^{-1} = (2 \, 1 \, 3) \quad \text{or equivalently } (1 \, 2 \, 3)^{-1} = (1 \, 3 \, 2)
\]
3. **Inverse of \(c\)**:
Similarly reversing the mappings gives us:
- \(1 \to 2\) gives \(2 \leftarrow 1\).
- \(2 \to 3\) gives \(3 \leftarrow 2\).
- \(3 \to 1\) gives \(1 \leftarrow 3\).
Thus, \(c^{-1}\):
\[
c^{-1} = (3 \, 2 \, 1) \quad \text{or } c^{-1} = (1 \, 3 \, 2)
\]
### c) Find the order of each permutation.
1. **Order of \(a\)**:
The identity permutation has order \(1\):
\[
\text{Order}(a) = 1
\]
2. **Order of \(b\)**:
The cycle \((1 \, 3 \, 2)\) has length \(3\):
\[
\text{Order}(b) = 3
\]
3. **Order of \(c\)**:
The cycle \((1 \, 2 \, 3)\) also has length \(3\):
\[
\text{Order}(c) = 3
\]
### d) Determine whether each permutation is even or odd.
1. **\(a\)**:
The identity permutation is even (0 transpositions):
\[
a \text{ is even.}
\]
2. **\(b\)**:
The cycle \((1 \, 3 \, 2)\) can be expressed as two transpositions \((1 \, 2)(2 \, 3)\), hence it is odd (an odd number of transpositions):
\[
b \text{ is odd.}
\]
3. **\(c\)**:
The cycle \((1 \, 2 \, 3)\) can similarly be expressed in 2 transpositions \((1 \, 2)(2 \, 3)\), hence it is odd:
\[
c \text{ is odd.}
\]
### Summary:
- **Cycle Notation**:
- \( a = e \)
- \( b = (1 \, 3 \, 2) \)
- \( c = (1 \, 2 \, 3) \)
- **Inverse**:
- \( a^{-1} = a \)
- \( b^{-1} = (2 \, 1 \, 3) \) or \(b^{-1} = (1 \, 3 \, 2)\)
- \( c^{-1} = (3 \, 2 \, 1) \) or \(c^{-1} = (1 \, 3 \, 2)\)
- **Order**:
- \( \text{Order}(a) = 1 \)
- \( \text{Order}(b) = 3 \)
- \( \text{Order}(c) = 3 \)
- **Parity**:
- \( a \text{ is even} \)
- \( b \text{ is odd} \)
- \( c \text{ is odd} \)
### a) Write these permutations in cycle notation.
Given permutations are:
1. **Permutation \(a\)**:
\[
a = \begin{pmatrix}
1 & 2 & 3 \\
1 & 2 & 3
\end{pmatrix}
\]
This means \(1 \to 1\), \(2 \to 2\), and \(3 \to 3\). In cycle notation, it is the identity permutation.
\[
a = (1)(2)(3) \quad \text{or simply } a = e
\]
2. **Permutation \(b\)**:
\[
b = \begin{pmatrix}
1 & 2 & 3 \\
3 & 1 & 2
\end{pmatrix}
\]
This means \(1 \to 3\), \(2 \to 1\), and \(3 \to 2\). In cycle notation, it can be written as:
\[
b = (1 \, 3 \, 2)
\]
3. **Permutation \(c\)**:
\[
c = \begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1
\end{pmatrix}
\]
This means \(1 \to 2\), \(2 \to 3\), and \(3 \to 1\). In cycle notation, it is:
\[
c = (1 \, 2 \, 3)
\]
### b) Find the inverse of each permutation.
1. **Inverse of \(a\)**:
The identity permutation is self-inverse:
\[
a^{-1} = a = e
\]
2. **Inverse of \(b\)**:
To find the inverse, we reverse the mappings:
- From \(2 \to 1\), we have \(1 \leftarrow 2\).
- From \(1 \to 3\), we have \(3 \leftarrow 1\).
- From \(3 \to 2\), we have \(2 \leftarrow 3\).
Thus, the inverse \(b^{-1}\) can be expressed as:
\[
b^{-1} = (2 \, 1 \, 3) \quad \text{or equivalently } (1 \, 2 \, 3)^{-1} = (1 \, 3 \, 2)
\]
3. **Inverse of \(c\)**:
Similarly reversing the mappings gives us:
- \(1 \to 2\) gives \(2 \leftarrow 1\).
- \(2 \to 3\) gives \(3 \leftarrow 2\).
- \(3 \to 1\) gives \(1 \leftarrow 3\).
Thus, \(c^{-1}\):
\[
c^{-1} = (3 \, 2 \, 1) \quad \text{or } c^{-1} = (1 \, 3 \, 2)
\]
### c) Find the order of each permutation.
1. **Order of \(a\)**:
The identity permutation has order \(1\):
\[
\text{Order}(a) = 1
\]
2. **Order of \(b\)**:
The cycle \((1 \, 3 \, 2)\) has length \(3\):
\[
\text{Order}(b) = 3
\]
3. **Order of \(c\)**:
The cycle \((1 \, 2 \, 3)\) also has length \(3\):
\[
\text{Order}(c) = 3
\]
### d) Determine whether each permutation is even or odd.
1. **\(a\)**:
The identity permutation is even (0 transpositions):
\[
a \text{ is even.}
\]
2. **\(b\)**:
The cycle \((1 \, 3 \, 2)\) can be expressed as two transpositions \((1 \, 2)(2 \, 3)\), hence it is odd (an odd number of transpositions):
\[
b \text{ is odd.}
\]
3. **\(c\)**:
The cycle \((1 \, 2 \, 3)\) can similarly be expressed in 2 transpositions \((1 \, 2)(2 \, 3)\), hence it is odd:
\[
c \text{ is odd.}
\]
### Summary:
- **Cycle Notation**:
- \( a = e \)
- \( b = (1 \, 3 \, 2) \)
- \( c = (1 \, 2 \, 3) \)
- **Inverse**:
- \( a^{-1} = a \)
- \( b^{-1} = (2 \, 1 \, 3) \) or \(b^{-1} = (1 \, 3 \, 2)\)
- \( c^{-1} = (3 \, 2 \, 1) \) or \(c^{-1} = (1 \, 3 \, 2)\)
- **Order**:
- \( \text{Order}(a) = 1 \)
- \( \text{Order}(b) = 3 \)
- \( \text{Order}(c) = 3 \)
- **Parity**:
- \( a \text{ is even} \)
- \( b \text{ is odd} \)
- \( c \text{ is odd} \)