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.