Explain why the set of prime numbers under multiplication is not a group

Question

I understand that if you multiply two prime numbers we don't get the inverse. I need help putting this into an argument.

Answer 1

A group is required to be closed. You specified multiplication as the operation. Closed under multiplication means that the product of any two elements of the group is necessarily another element. However, by definition, if a number is the product of any two primes, it is not in the set of primes. Therefore, it is not closed. In fact, primes cannot be a group under any function you would see in high school. This is because there is no function which can generate exclusively prime numbers. If the primes were closed under a function, you could keep on combining them to get larger and larger primes, which violates the rule about no generating function.

Answer 2

It is not about inverses. The product of two primes is never a prime, so you don't even have an operation.