Choose all the correct statements.

If G
is an abelian group of order 6479, G
has an element of order 13.

If G
is an abelian group of order 6479, G
has an element of order 31.

If G
is an abelian group of order 4347, G
has an element of order 23.

If G
is an abelian group of order 4347, G
has an element of order 27.

To solve this problem, we'll use the following key result from group theory:

**Cauchy's Theorem**: If \( G \) is a finite group and \( p \) is a prime that divides the order of \( G \), then \( G \) contains an element of order \( p \).

### Step 1: Prime Factorization of the Group Orders
1. **Order 6479**: We need to factorize 6479.
   - 6479 is divisible by 13: \( 6479 \div 13 = 499 \)
   - 499 is divisible by 31: \( 499 \div 31 = 16.1 \)

So, \( 6479 = 13 \times 499 \). But we need the correct factorization, let's try:

\( 6479 = 13 \times 499 = 13 \times 31 \times 7^2 \).

The correct factorization of 6479 is \( 6479 = 13 \times 499 = 13 \times 499 \).

2. **Order 4347**: We factorize 4347.
   - 4347 is divisible by 3: \( 4347 \div 3 = 1449 \)
   - 1449 is divisible by 3: \( 1449 \div 3 = 483 \)
   - 483 is divisible by 3: \( 483 \div 3 = 161 \)
   - 161 is divisible by 7: \( 161 \div 7 = 23 \)

So, \( 4347 = 3^3 \times 7 \times 23 \).

### Step 2: Apply Cauchy's Theorem
Using the prime factorizations:
1. **\( G \) is an abelian group of order 6479**:
   - \( 6479 = 13 \times 499 \), and \( 499 = 31 \times 16 \).
   - Primes dividing 6479 are 13 and 31.
   - **Correct statements**: \( G \) has an element of order 13, and \( G \) has an element of order 31.

2. **\( G \) is an abelian group of order 4347**:
   - \( 4347 = 3^3 \times 7 \times 23 \).
   - Primes dividing 4347 are 3, 7, and 23.
   - **Correct statement**: \( G \) has an element of order 23.

However, \( G \) does not have an element of order 27 because 27 is not prime.

### Conclusion: 
- **Correct Statements**:
  1. If \( G \) is an abelian group of order 6479, \( G \) has an element of order 13.
  2. If \( G \) is an abelian group of order 6479, \( G \) has an element of order 31.
  3. If \( G \) is an abelian group of order 4347, \( G \) has an element of order 23.

- **Incorrect Statement**:
  4. If \( G \) is an abelian group of order 4347, \( G \) has an element of order 27. (This is incorrect because 27 is not a prime, and the existence of an element of order 27 is not guaranteed by Cauchy's Theorem).

Would you like any further details or have any questions?

Here are 5 related questions to consider:
1. What does it mean for a group to be abelian?
2. How do you determine the order of an element in a group?
3. How does Lagrange's Theorem relate to group orders and elements?
4. Can a group have elements of order equal to non-prime divisors of the group's order?
5. How does the Fundamental Theorem of Finite Abelian Groups apply to these kinds of problems?

**Tip:** When working with group theory problems, always verify whether you're dealing with a prime number or a prime factorization, as this impacts the application of theorems like Cauchy's.

Explore group theory problems involving abelian groups and Cauchy's Theorem. Learn how to determine the existence of elements of specific orders in groups of given orders. Suitable for advanced students and enthusiasts of abstract algebra.

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