Let's analyze each statement one by one to determine which are correct.

1. Statement: "Если разложение группы $G$ на левые и правые смежные классы по подгруппе $H$ совпадает, то $H$ — нормальная подгруппа."

   • Explanation: This statement is indeed correct. If the left and right cosets of a subgroup $H$ in a group $G$ coincide, then $H$ is a normal subgroup of $G$.
   • Conclusion: This statement is true.

2. Statement: "Группы $\mathbb{Z}_3 \times \mathbb{Z}_{15}$ и $\mathbb{Z}_9 \times \mathbb{Z}_5$ изоморфны."

   • Explanation: To determine if these groups are isomorphic, we need to examine their structure. The group $\mathbb{Z}_3 \times \mathbb{Z}_{15}$ is isomorphic to $\mathbb{Z}_{45}$ because $3$ and $15$ are coprime, so the direct product will have order $3 \times 15 = 45$. For $\mathbb{Z}_9 \times \mathbb{Z}_5$, $9$ and $5$ are also coprime, giving a group isomorphic to $\mathbb{Z}_{45}$. Therefore, these two groups are isomorphic.
   • Conclusion: This statement is true.

3. Statement: "Индекс подгруппы порожденной элементами $(1\ 2), (2\ 3)$ в группе $S_4$ равен 2."

   • Explanation: The elements $(1\ 2)$ and $(2\ 3)$ in the symmetric group $S_4$ generate the subgroup $A_4$, which is the alternating group of order 12. The order of $S_4$ is 24, so the index of $A_4$ in $S_4$ is $\frac{24}{12} = 2$.
   • Conclusion: This statement is true.

4. Statement: "В группе $S_4$ есть подгруппа $H$ порядка 9."

   • Explanation: The group $S_4$ has order 24. Any subgroup of $S_4$ must have an order that divides 24 by Lagrange's theorem. Since 9 does not divide 24, there cannot be a subgroup of order 9 in $S_4$.
   • Conclusion: This statement is false.

Summary of Correct Statements

The correct statements are:

• The first statement (about normal subgroups).
• The second statement (isomorphism of groups).
• The third statement (index of a subgroup in $S_4$).

Would you like further explanation on any of these points?

Additional Questions

1. What are the general criteria for determining if two groups are isomorphic?
2. How does Lagrange's theorem help in identifying possible subgroup orders?
3. What properties of cosets help identify normal subgroups?
4. Can you explore more examples of normal and non-normal subgroups in various groups?
5. How does the structure of the symmetric group $S_n$ influence the possible subgroups and their orders?

Tip

When determining if two groups are isomorphic, consider their orders, the structure of their elements, and possible decompositions, such as direct products with coprime orders.