I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem.(adsbygoogle = window.adsbygoogle || []).push({});

I'm learning group theory on my own, 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) 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"?

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# Beginning Group Theory, wondering if subset of nat numbers are groups?

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