Here is a translation and breakdown of the mathematical questions from the image you uploaded:

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.