https://file.aiquickdraw.com/mathbot/file/1733391291625cq807osk.jpg

Here is the analysis of the statements in the image:

1. **"В группе \( S_4 \) есть подгруппа \( H \) порядка 5."**  
   The group \( S_4 \) (symmetric group of degree 4) has order \( 4! = 24 \). A subgroup of order 5 would require that 5 divides 24, which is not true.  
   **This statement is false.**

2. **"Индекс подгруппы \( A_n \) в группе \( S_n \) равен 2."**  
   The alternating group \( A_n \) consists of all even permutations of \( S_n \), and its order is \( |A_n| = \frac{|S_n|}{2} \). Therefore, the index of \( A_n \) in \( S_n \) is indeed 2.  
   **This statement is true.**

3. **"Центр группы всегда конечен."**  
   The center of a group (set of elements commuting with all other elements) is not always finite. For example, the center of an infinite group like \( \mathbb{R}^n \) (under addition) is infinite.  
   **This statement is false.**

4. **"Группы \( \mathbb{Z}_3 \times \mathbb{Z}_{10} \) и \( \mathbb{Z}_{15} \times \mathbb{Z}_2 \) изоморфны."**  
   To check isomorphism, we analyze their structures:  
   - \( \mathbb{Z}_3 \times \mathbb{Z}_{10} \): Use the Chinese Remainder Theorem. Since \( \text{gcd}(3, 10) = 1 \), this is isomorphic to \( \mathbb{Z}_{30} \).  
   - \( \mathbb{Z}_{15} \times \mathbb{Z}_2 \): Similarly, since \( \text{gcd}(15, 2) = 1 \), this is also isomorphic to \( \mathbb{Z}_{30} \).  
   Thus, the two groups are isomorphic.  
   **This statement is true.**

### Final Answers:
- True statements:  
  - "Индекс подгруппы \( A_n \) в группе \( S_n \) равен 2."
  - "Группы \( \mathbb{Z}_3 \times \mathbb{Z}_{10} \) и \( \mathbb{Z}_{15} \times \mathbb{Z}_2 \) изоморфны."

Let me know if you'd like a deeper explanation of any part!  

### Related Questions:
1. How is the order of a subgroup determined in a symmetric group?
2. Can we generalize the index of \( A_n \) in \( S_n \) to higher \( n \)?
3. What examples exist of infinite groups with infinite centers?
4. How does the Chinese Remainder Theorem help in determining group isomorphism?
5. Why can't a group of order 24 have a subgroup of order 5?  

### Tip:  
When analyzing group structures, always verify divisibility rules and the order of elements to rule out false subgroup claims.

Укажите все верные утверждения из предложенного списка: 
1. В группе S4 есть подгруппа H порядка 5. 
2. Индекс подгруппы An в группе Sn равен 2. 
3. Центр группы всегда конечен. 
4. Группы Z3 × Z10 и Z15 × Z2 изоморфны.

Explore the analysis of subgroups in the symmetric group S4, determine the index of An in Sn, discuss the finiteness of group centers, and verify the isomorphism between Z3 × Z10 and Z15 × Z2 using key concepts from group theory.

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