The set of prime numbers under multiplication 

Let's determine whether the set of prime numbers under multiplication forms a group. We will check each of the four properties required for a set to be a group:

### 1. **Closure**
   - The set of prime numbers includes numbers like \(2, 3, 5, 7, 11, \) and so on.
   - **Under multiplication:** If you take any two prime numbers and multiply them, the result is **not** a prime number (e.g., \(2 \times 3 = 6\), \(5 \times 7 = 35\)).
   - **Conclusion:** The set of prime numbers is **not closed** under multiplication because the product of two prime numbers is not necessarily a prime number.

### 2. **Associativity**
   - **Multiplication** is an associative operation, meaning that for any three numbers \(a\), \(b\), and \(c\), \((a \times b) \times c = a \times (b \times c)\).
   - **Conclusion:** This property would hold if the set was closed under multiplication, but since closure fails, we don’t need to verify further.

### 3. **Identity Element**
   - The identity element for multiplication is \(1\) because any number multiplied by \(1\) remains unchanged.
   - However, \(1\) is **not a prime number**.
   - **Conclusion:** The set of prime numbers does **not** contain an identity element under multiplication.

### 4. **Inverse Element**
   - For any prime number \(p\), there should exist another prime number \(q\) such that \(p \times q = 1\).
   - However, no prime number \(p\) can be multiplied by another prime number to give \(1\).
   - **Conclusion:** There are **no inverses** within the set of prime numbers under multiplication.

### Final Conclusion:
The set of prime numbers under multiplication **does not form a group** because it fails the properties of closure, identity element, and inverse element.

Would you like more details or have any further questions?

Here are five questions related to this topic:

1. What is an example of a set that forms a group under multiplication?
2. How does the failure of the closure property impact group formation?
3. Can a set of prime numbers form a group under any operation?
4. What are the implications of the absence of an identity element in a set?
5. How can we determine the inverse elements in a group?

**Tip:** Understanding the importance of the identity and inverse elements is crucial in group theory, as they are fundamental to the structure and behavior of groups.

Explore whether prime numbers can form a group under multiplication. Learn about the closure property, identity element, and inverse element in group theory. Discover why prime numbers do not satisfy the requirements to form a group.

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