14. Using cycle notation, list the elements in $$ D_{5} $$. What are $$ r $$ and $$ s $$ ? Write every element as a product of $$ r $$ and $$ s $$.

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To solve the problem, we first need to understand the dihedral group $$ D_5 $$, which represents the symmetries of a regular pentagon. This group consists of rotations and reflections.

### Step 1: Identify the elements of $$ D_5 $$

The dihedral group $$ D_5 $$ has 10 elements, which can be represented as follows:

- **Rotations**:
  - $$ e $$ (the identity rotation)
  - $$ r $$ (rotation by $$ 72^\circ $$)
  - $$ r^2 $$ (rotation by $$ 144^\circ $$)
  - $$ r^3 $$ (rotation by $$ 216^\circ $$)
  - $$ r^4 $$ (rotation by $$ 288^\circ $$)

- **Reflections**:
  - $$ s $$ (reflection across a line through a vertex)
  - $$ sr $$ (reflection after rotating by $$ 72^\circ $$)
  - $$ sr^2 $$ (reflection after rotating by $$ 144^\circ $$)
  - $$ sr^3 $$ (reflection after rotating by $$ 216^\circ $$)
  - $$ sr^4 $$ (reflection after rotating by $$ 288^\circ $$)

### Step 2: Define $$ r $$ and $$ s $$

- $$ r $$ is the rotation by $$ 72^\circ $$.
- $$ s $$ is a reflection across a line through one of the vertices of the pentagon.

### Step 3: Write every element as a product of $$ r $$ and $$ s $$

Now, we can express each element of $$ D_5 $$ as a product of $$ r $$ and $$ s $$:

1. $$ e = r^0 $$
2. $$ r = r^1 $$
3. $$ r^2 = r^2 $$
4. $$ r^3 = r^3 $$
5. $$ r^4 = r^4 $$
6. $$ s = s $$
7. $$ sr = sr^1 $$
8. $$ sr^2 = sr^2 $$
9. $$ sr^3 = sr^3 $$
10. $$ sr^4 = sr^4 $$

### Summary

The elements of $$ D_5 $$ in cycle notation are:
- $$ e, r, r^2, r^3, r^4, s, sr, sr^2, sr^3, sr^4 $$

Where:
- $$ r $$ is the rotation by $$ 72^\circ $$.
- $$ s $$ is a reflection.

Each element can be expressed as a product of $$ r $$ and $$ s $$ as shown above.
The elements of $$ D_5 $$ are: - $$ e, r, r^2, r^3, r^4, s, sr, sr^2, sr^3, sr^4 $$ Here, $$ r $$ is a rotation by $$ 72^\circ $$ and $$ s $$ is a reflection. Each element can be written as a product of $$ r $$ and $$ s $$.

Using cycle notation, list the elements in . What are and ? Write every
element as a product of and .

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Solución

Step 1: Identify the elements of

Step 2: Define and

Step 3: Write every element as a product of and

Summary

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Using cycle notation, list the elements in . What are and ? Write every element as a product of and .