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

1. Aug 22, 2014

### mathacka

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.

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"?

2. Aug 22, 2014

### HallsofIvy

Staff Emeritus

3. Aug 23, 2014

### xAxis

It's been some time since I've done any abstract algebra, but this:
QUOTE:
Does this break down because a function might not necessarily be considered a "binary operation"?
is definitely not true. The whole point of groups is to find a structure that might be considered to have properties of the group. Your function is clearly a binary operation, and that's all you need to know that it will be a group if it satisfies other properties.