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1. **Give an example of a group in which all non-identity elements have order 2. Construct its Cayley table.**
   
2. **Describe the orders of all elements in the group of residues \( \mathbb{Z}_6 \) under addition.**

3. **Prove that the group \( (\mathbb{Z}_n, +) \), the group of integers modulo \( n \), is isomorphic to the cyclic group \( C_n \).**

4. **Does a non-commutative group exist with 2 elements? What about with 3 elements? Or with \( p \) elements, where \( p \) is a prime number?**

5. **Consider \( (\mathbb{R}, *) \) with multiplication defined as \( a * b = a^b \). Check its properties: will this structure be a ring? A field?**

6. **Consider \( \mathbb{Z} \) along with GCD or LCM, and check its properties. Does a non-commutative group structure exist on \( \mathbb{Z} \)?**

7. **Define a ring structure on 4 elements. Define a field structure on 4 elements.**

Would you like to focus on any specific question first or get detailed answers for multiple? Here are 5 related questions that expand on this topic:

1. How do we construct a Cayley table for a group?
2. What are the generators of cyclic groups like \( C_n \)?
3. What is an example of a non-commutative group?
4. How do GCD and LCM operations define group structures in \( \mathbb{Z} \)?
5. What are the key properties that distinguish rings from fields?

**Tip**: When working with Cayley tables, it helps to focus on understanding the symmetry of the group's structure.

1. Приведите пример группы, в которой все неединичные элементы имеют порядок 2. Постройте ее таблицу Кэли.
2. Опишите порядки всех элементов в группе остатков Z6 со сложением.
3. Докажите, что группа (Zn, +) как группа остатков целых чисел по модулю n изоморфна циклической группе Cn.
4. Существует ли некоммутативная группа на 2 элементах? А на 3 элементах? А на p элементах, где p — простое число?
5. Рассмотрите (R, *) с умножением a * b = a^b. Проверьте его свойства: будет ли эта структура кольцом? А полем?
6. Рассмотрите Z вместе с НОД или НОК, проверьте свойства. Существует ли некоммутативная структура группы на Z?
7. Задайте структуру кольца на 4 элементах. Задайте структуру поля на 4 элементах.

Delve into advanced group theory with questions about cyclic groups, Cayley tables, non-commutative groups, and structures of rings and fields. Understand the isomorphism between Zn and Cn, and explore the group properties under addition and multiplication.

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