3. Write out Cayley tables for groups formed by the symmetries of a rectangle and for $$ \left(\mathbb{Z}_{4 y},+\right) $$. How many elements are in each group? Are the groups the same? Why or why not?

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To solve this problem, we will first identify the groups formed by the symmetries of a rectangle and the group $$(\mathbb{Z}_{4}, +)$$. Then, we will construct the Cayley tables for both groups and analyze their properties.

### Step 1: Symmetries of a Rectangle

The symmetries of a rectangle include:
1. Identity (no change)
2. Rotation by 180 degrees
3. Reflection across the vertical axis
4. Reflection across the horizontal axis

Thus, the group of symmetries of a rectangle has 4 elements. We can denote these symmetries as follows:
- $$ e $$: Identity
- $$ r $$: Rotation by 180 degrees
- $$ v $$: Reflection across the vertical axis
- $$ h $$: Reflection across the horizontal axis

### Step 2: Cayley Table for the Symmetries of a Rectangle

Now, we will construct the Cayley table for the group of symmetries of a rectangle. The operation is function composition.

$$
\begin{array}{c|cccc}
\cdot & e & r & v & h \
\hline
e & e & r & v & h \
r & r & e & h & v \
v & v & h & e & r \
h & h & v & r & e \
\end{array}
$$

### Step 3: Group $$(\mathbb{Z}_{4}, +)$$

The group $$(\mathbb{Z}_{4}, +)$$ consists of the integers modulo 4 under addition. The elements are:
- $$ 0 $$
- $$ 1 $$
- $$ 2 $$
- $$ 3 $$

### Step 4: Cayley Table for $$(\mathbb{Z}_{4}, +)$$

Now, we will construct the Cayley table for $$(\mathbb{Z}_{4}, +)$$.

$$
\begin{array}{c|cccc}
+ & 0 & 1 & 2 & 3 \
\hline
0 & 0 & 1 & 2 & 3 \
1 & 1 & 2 & 3 & 0 \
2 & 2 & 3 & 0 & 1 \
3 & 3 & 0 & 1 & 2 \
\end{array}
$$

### Step 5: Analysis of the Groups

1. **Number of Elements**:
   - The group of symmetries of a rectangle has 4 elements.
   - The group $$(\mathbb{Z}_{4}, +)$$ also has 4 elements.

2. **Are the Groups the Same?**:
   - Although both groups have the same number of elements, they are not the same group. The group of symmetries of a rectangle is non-abelian (the order of operations matters), while $$(\mathbb{Z}_{4}, +)$$ is abelian (the order of addition does not matter).
   - The structure of the operations is different: the symmetries involve geometric transformations, while $$(\mathbb{Z}_{4}, +)$$ involves arithmetic addition.

### Conclusion

In summary, both groups have 4 elements, but they are not the same due to the nature of their operations (non-abelian vs. abelian).
The group of symmetries of a rectangle and $$(\mathbb{Z}_{4}, +)$$ both have 4 elements but are different because one is non-abelian (symmetries involve geometric transformations) and the other is abelian (involves arithmetic addition).

3. Write out Cayley tables for groups formed by the symmetries of a rectangle and for \( \left(\mathbb{Z}_{4 y},+\right) \). How many elements are in each group? Are the groups the same? Why or why not?

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