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.