Let's break this problem into **two steps**, providing explanations at each stage, and then give final answers to whether \( G \) is abelian and whether \( G \) is finite. --- ### **Step 1: List and Describe the Elements of \( G \)** #### **Analysis** The pattern consists of an infinite row of hearts, each equally spaced. We are asked about the group of symmetries of this infinite strip. For such infinite, regularly repeating patterns, the classic symmetries are: - **Translations**: Shift the whole pattern left or right by any integer number of spaces. - **Reflections**: Flip the pattern over a line of symmetry. Here, any line passing through a heart or halfway between two hearts is a line of symmetry. #### **Notation** - Let \( t_n \) denote the translation symmetry that shifts every heart by \( n \) spaces, where \( n \in \mathbb{Z} \). - Let \( r_a \) denote the reflection symmetry over the line at position \( a \), where \( a \) can be any integer (through a heart) or half-integer (between hearts). So, the full list of elements in \( G \) is: - All translations: \( \{ t_n : n \in \mathbb{Z} \} \) - All reflections: \( \{ r_a : a \in \mathbb{Z} \text{ or } a \in \mathbb{Z} + \frac{1}{2} \} \) **Explanation:** We list translations and reflections because those are the only isometries (distance-preserving symmetries) that map the infinite row of hearts to itself. --- ### **Step 2: Describe the Group Operation and Properties** #### **Group Operation** - **Translation + Translation:** \( t_m \circ t_n = t_{m+n} \) - **Reflection + Reflection:** \( r_a \circ r_b = t_{2(a-b)} \) (composition of two reflections over parallel lines is a translation by twice the distance between the lines) - **Translation + Reflection:** \( t_n \circ r_a = r_{a+n} \) (translation followed by reflection is reflection over the shifted line) - **Reflection + Translation:** \( r_a \circ t_n = r_{a-n} \) (reflection followed by translation is reflection over the shifted line in the opposite direction) #### **Is \( G \) Abelian?** Check commutativity: - \( t_m \circ t_n = t_n \circ t_m \) (translations commute) - \( r_a \circ r_b = t_{2(a-b)} \), but \( r_b \circ r_a = t_{2(b-a)} = t_{-2(a-b)} \) (so, does **not** commute in general) - \( t_n \circ r_a = r_{a+n} \), but \( r_a \circ t_n = r_{a-n} \) (**not** the same) So, **\( G \) is not abelian**. #### **Is \( G \) Finite?** There are infinitely many translations (\( n \in \mathbb{Z} \)), and infinitely many possible axes for reflections (\( a \in \mathbb{Z} \) or \( \mathbb{Z} + 1/2 \)). **So, \( G \) is infinite.** --- ## **Final Answers** 1. **List of Elements:** - All translations by integer steps (\( t_n \)), and all reflections over axes at integer or half-integer positions (\( r_a \)). 2. **Group Operation:** - Given above for each pair of types. 3. **Is \( G \) Abelian?** - **No**, it is not abelian. 4. **Is \( G \) Finite?** - **No**, it is infinite.