We need to analyze the symmetries of an $8 \times 10$ white piece of paper that preserve its position on the table and determine how many elements in $G \times \mathbb{Z}_3$ have order 3.

Step 1: Understanding $G$

The group $G$ consists of the symmetries of a rectangle that leave it in the same place on the table. The allowed symmetries are:

• The identity transformation $e$.
• $180^\circ$ rotation ($r$).
• Reflections across the vertical axis ($s_v$), the horizontal axis ($s_h$), and the two diagonals ($s_d$ and $s_{d'}$).

The dihedral group $D_2$ (or $D_4$ in general notation) describes these symmetries, giving us 8 elements: $G = \{ e, r, s_v, s_h, s_d, s_{d'}, s_{vh}, s_{dd'} \}$

Step 2: Understanding $\mathbb{Z}_3$

The group $\mathbb{Z}_3$ consists of three elements: $\mathbb{Z}_3 = \{0, 1, 2\}$ where each number represents multiplication by a primitive 3rd root of unity.

Step 3: Elements of Order 3 in $G \times \mathbb{Z}_3$

An element $(g, z) \in G \times \mathbb{Z}_3$ has order 3 if and only if: $(g, z)^3 = (e, 0)$

This means that:

1. $g^3 = e$ in $G$.
2. $z$ must be a generator of $\mathbb{Z}_3$, meaning $z = 1$ or $z = 2$.

Since $G$ has no elements of order 3 (as all elements square to identity or have order 2), the only way to achieve order 3 in $G \times \mathbb{Z}_3$ is by picking $g = e$ (identity element), while $z$ is an element of order 3.

Step 4: Counting the Elements

Since $G$ has only one element of order 1 (identity $e$), the valid elements of order 3 in $G \times \mathbb{Z}_3$ are: $(e, 1), (e, 2)$

Thus, there are 2 elements of order 3.

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Do you want a more detailed explanation or have any questions?

Here are 5 related questions:

1. What are the properties of the dihedral group $D_4$ in general?
2. How do we determine the order of an element in a direct product of groups?
3. What would happen if the paper were a square instead of a rectangle?
4. How does the order of elements in $\mathbb{Z}_n$ affect direct product group structures?
5. What changes if we consider symmetries that allow flipping the paper over?

Tip: Always check the order of elements in individual groups before analyzing direct products.