Math 360, Fall 2021, Assignment 11

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

Read:

1. Section 9.

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

1. Fixed point (of a permutation $\pi$).
2. Moving point (of a permutation $\pi$).
3. Disjoint (permutations $\pi$ and $\sigma$).
4. Orbit (of a permutation $\pi$).
5. Cycle.
6. $(i_1,\dots,i_k)$ (the cycle determined by the sequence $i_1,\dots,i_k$).
7. Length (of a cycle).

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

1. Theorem relating $\sigma\tau$ to $\tau\sigma$, when $\sigma$ and $\tau$ are disjoint.
2. Theorem concerning disjoint cycle decomposition.

Carefully practice the following calculations, giving a worked example of each:

1. Conversion of two-row notation to cycle notation.
2. Conversion of cycle notation to two-row notation.
3. Composition of two permutations, expressed in cycle notation.
4. Inversion of a permutation, expressed in cycle notation.

Solve the following problems:

1. Section 9, problems 1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. (Hint for 14-16: recall that the order of a finite group element $g$ is the least positive integer $n$ with $g^n=1$. For a permutation, this will be the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.)

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Definitions:

1. Fixed point (of a permutation $\pi$): $i$ is said to be a fixed point of $\pi$ if $\pi (i) = I$.
2. Moving point (of a permutation $\pi$): $i$ is said to be a moving point of $\pi$ if not $\pi (i) = i$.

Theorems:

1. Theorem relating $\sigma\tau$ to $\tau\sigma$, when $\sigma$ and $\tau$ are disjoint.
2. Theorem concerning disjoint cycle decomposition.

Calculation Problems:

1. Conversion of two-row notation to cycle notation.
2. Conversion of cycle notation to two-row notation.
3. Composition of two permutations, expressed in cycle notation.
4. Inversion of a permutation, expressed in cycle notation.

Book Problems:

1.

3.

5.

7.

9.

10.

11.

13.

14.

15.

16.