Let's go through each statement to determine its validity.

1. The groups $\mathbb{Z}_3 \times \mathbb{Z}_{10}$ and $\mathbb{Z}_{15} \times \mathbb{Z}_2$ are isomorphic.

   • To determine if two groups are isomorphic, their structures, specifically their orders, must align.
   • The group $\mathbb{Z}_3 \times \mathbb{Z}_{10}$ has elements of orders 3 and 10, respectively, and $3$ and $10$ are coprime, so the order of this group is $3 \times 10 = 30$.
   • The group $\mathbb{Z}_{15} \times \mathbb{Z}_2$ has orders 15 and 2, also coprime, so the order of this group is $15 \times 2 = 30$.
   • Since both groups have the same order and are direct products of cyclic groups whose orders are pairwise coprime, they are indeed isomorphic.
   • This statement is true.

2. The index of the subgroup $A_n$ in the group $S_n$ is equal to 2.

   • The alternating group $A_n$ consists of all even permutations in $S_n$.
   • Since $A_n$ contains half of the elements of $S_n$, the index of $A_n$ in $S_n$ is indeed 2.
   • This statement is true.

3. In the group $S_4$, there exists a subgroup $H$ of order 5.

   • The order of $S_4$ is 24. By Lagrange's theorem, any subgroup $H$ of $S_4$ must have an order that divides 24.
   • The possible orders of subgroups of $S_4$ are therefore divisors of 24: $1, 2, 3, 4, 6, 8, 12,$ and $24$.
   • 5 is not a divisor of 24, so there cannot be a subgroup of order 5 in $S_4$.
   • This statement is false.

4. The center of a group is always finite.

   • The center of a group $G$, denoted $Z(G)$, is the set of elements that commute with all elements in $G$.
   • There exist groups with infinite centers; for example, the center of an infinite abelian group is the group itself, which may be infinite.
   • This statement is false.

Summary of Correct Statements

The correct statements are:

• Statement 1: The groups $\mathbb{Z}_3 \times \mathbb{Z}_{10}$ and $\mathbb{Z}_{15} \times \mathbb{Z}_2$ are isomorphic.
• Statement 2: The index of the subgroup $A_n$ in the group $S_n$ is equal to 2.

Would you like further clarification on any of these points?

Here are some follow-up questions for further exploration:

1. What are the criteria for two direct products of cyclic groups to be isomorphic?
2. How does one calculate the index of a subgroup in general?
3. Why can’t a subgroup of order 5 exist in $S_4$?
4. What are examples of groups with infinite centers?
5. How does the structure of $\mathbb{Z}_n$ influence the isomorphism of direct products?

Tip: When determining the isomorphism of two groups, check if their orders and structural properties (like being a direct product of coprime orders) align.