Math 360, Fall 2013, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Carefully define the following terms, then give one example and one non-example of each:

1. Orbits (of a permutation).
2. Cycle.
3. Disjoint cycles.
4. Transposition.
5. Even permutation.
6. Odd permutation.
7. Alternating group on $n$ letters.

Carefully state the following theorems (you need not prove them):

1. Theorem concerning generation of $S_n$ by transpositions (Corollary 9.12 in the text).
2. Theorem concerning the parity of a permutation (Theorem 9.15).
3. Theorem concerning the order of the alternating group (Theorem 9.20).

Solve the following problems:

1. Section 9, problems 3, 5, 9, 10, 15, 24, and 29.

--------------------End of assignment--------------------

Questions:

Book Problems:

1. 9.3

   Find all orbits of:

$$ \left ( \begin{array}{cccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2 & 3 & 5 & 1 & 4 & 6 & 8 & 7 \end{array}\right ) $$

1. 9.5

   Find all orbits of:

$$ \sigma :\mathbb{Z}\rightarrow \mathbb{Z} $$

where \(\sigma(n)=n+2\).

1. 9.9

   Compute the product: \((1,2)(4,7,8),7,2,8,1,5\).

2. 9.10

   Express the permutation as a product of disjoint cycles:

$$ \left ( \begin{array}{cccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 8 & 2 & 6 & 3 & 7 & 4 & 5 & 1 \end{array} \right ) $$

1. 9.15

   Find the maximum possible order of an element of \(S_6\).

2. 9.24

   Which of the permutations in \(S_3\) are even permutations? Give the table for the alternating group \(A_3\).

3. 9.29

   Show that for every subgroup \(H\) of \(S_n\) for \(n\geq 2\), either all the permutations in \(H\) are even or exactly half of them are even.

Solutions:

Definitions:

1. Orbits of a Permutation.

   Definition:

   Let \(\sigma\) be a permutation of \(A\). We define the equivalence relation \(~\) by:$$\forall a,b\in A: a~b \leftrightarrow b = \sigma^n(a)$$

   These equivalence classes are the orbits of \(\sigma\).

   Example:

   Non-Example:

2. Cycle.

   Definition:

   Example:

   Non-Example:

3. Disjoint Cycles.

   Definition:

   Example:

   Non-Example:

4. Transposition.

   Definition:

   Example:

   Non-Example:

5. Even Permutation.

   Definition:

   Example:

   Non-Example:

6. Odd Permutation.

   Definition:

   Example:

   Non-Example:

7. Alternating Group on \(n\) Letters.

   Definition:

   Example:

   Non-Example:

Theorems:

1. Theorem Concerning Generation of \(S_n\) by Transpositions.

2. Theorem Concerning the Parity of a Permutation.

3. Theorem Concerning the Order of the Alternating Group

Book Problems:

1. 9.3

2. 9.5

3. 9.9

4. 9.10

5. 9.15

6. 9.24

7. 9.29