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

[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"?

 

[HallsofIvy]

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 *)?

Can "the inputs x and y" what you seem to have left out the predicate of this sentence.

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)?

?? "(x, y)-> f(x,y)" is NOT a set. Are you asking if the set of natural numbers, with the operation "x+ y"= f(x, y)= xp*yq= (xy)pq for fixed primes p and q. (Did you really mean "*"? This problem seems more interesting if we define the operation to be xp+ yq.)

Does this break down because a function might not necessarily be considered a "binary operation"?

IF you mean what I suggested above, this clearly is a "binary function"- it is applied to x and y.

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.

In order to be a group the operation must be associative. Is that true here?

Assuming you have shown that, the identity must be an integer, e, such that, for any integer x, f(x,e)= (xe)(pq)= x. What if x= 0?

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"?

By "prime number set" do you mean a set of prime numbers or a set containing a prime number of elements?

 

[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.