Let's analyze each statement to determine its correctness:

1. Центр группы всегда конечен.

   • This statement is not always true. The center of a group is finite if the group is finite, but in infinite groups, the center can also be infinite.

2. В группе $S_4$ есть подгруппа $H$ порядка 5.

   • This is incorrect. The order of $S_4$ is 24, and there is no subgroup of order 5 in $S_4$ since 5 does not divide 24 (by Lagrange's theorem).

3. Индекс подгруппы $A_n$ в группе $S_n$ равен 2.

   • This is correct. The alternating group $A_n$ has index 2 in $S_n$ because it consists of half the elements (those that are even permutations).

4. Группы $\mathbb{Z}_3 \times \mathbb{Z}_{10}$ и $\mathbb{Z}_{15} \times \mathbb{Z}_2$ изоморфны.

   • This is incorrect. $\mathbb{Z}_3 \times \mathbb{Z}_{10}$ and $\mathbb{Z}_{15} \times \mathbb{Z}_2$ have different structures because the orders of elements and the decomposition into cyclic groups differ.

Thus, the correct answer is:

• Only the third statement is true: Индекс подгруппы $A_n$ в группе $S_n$ равен 2.

Would you like a deeper dive into group theory concepts or have any questions?

Here are five related questions to consider:

1. What are some properties of the center of a group in different types of groups?
2. Can you give examples of subgroups in $S_n$ for various orders?
3. How does Lagrange's theorem help in identifying possible subgroup orders?
4. What are the conditions for two direct product groups to be isomorphic?
5. How does the structure of alternating and symmetric groups affect their subgroup relations?

Tip: When analyzing group properties, always consider Lagrange's theorem for finite groups and the group's decomposition for structure comparison.