I'm learning group theory and I'm trying to consider the "symmetry" of a certain group of natural numbers:
Here's the idea, all natural numbers are comprised of multiples of primes. So a subset would be those comprised of 2 unique primes.
Consider the function:
f(x,y) = xp * yq
where p and q are chosen unique prime numbers and x and y are natural numbers.
can the inputs x and y, with the function itself being thought of as "the operator", (since x*p and y*q is also a condition of the operator *)?
QUESTION 1: My main question is, can the set of (x,y) -> f(x,y) = xp * yq be thought of as a group? If so, what is the inverse function and identity (x,y)?
Does this break down because a function might not necessarily be considered a "binary operation"?
I have a feeling this is much simpler than what I'm making it out to be, my idea is to see what kind of symmetry or other symmetry-like structure(s) that prime numbers have on the generation of the natural numbers.
QUESTION 2: If this isn't a group in group theory, what kind of abstract algebraic structures ought I be looking at?
QUESTION 3: As a side note, can a finite set of 1 - n | (1, 2, 3, n) have a prime number set of "generators"? Are these primes technically group generators, or are they something else entirely different, perhaps named a "set generator"?