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

2. 9.10

3. 9.15

4. 9.24

5. 9.29

Solutions:

Definitions:

1. Orbits of a Permutation.

   Definition:

   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