Symmetries: Why Number Theory is Kind of Important

During my time as a physics student, I have often heard others in the department lament the fact that they have to take certain proof based upper division math courses for an elective credit. Among those, number theory is a popular course that physics students can take to fulfill this.

However, many don’t find it particularly enjoyable; a lot of physics students are concerned with the utility of a mathematical idea, which makes sense given the discipline, & are convinced number theory has no use in application. I think this shortsightedness is unfortunate, but easily correctable: just shed some light on a common application of ideas from number theory that are prominently used in physics! That application? Group theory, which will be the content of this post.

What is a Group?

Group theory is a fancy way of describing the analysis of symmetries. A symmetry, in the mathematical sense, refers to a system which remains invariant under certain transformations. These could be geometric objects, functions, sets, or any other type of system – physical or otherwise.

Physicists typically don’t need to be told how group theory is useful, they already know this to be the case. But a lot of undergrads are not able to properly study foundational aspects of groups because the degree plan doesn’t emphasize it, even though it underpins some of the most important physical laws – such as conservation principles – that we do, in fact, learn.

The easiest way to think of groups is to think of a set with an operation applied. For this set to indeed be a group, it must fulfill certain properties:

  • It must have a binary operation
  • It must be associative
  • It must contain an identity element
  • It must contain an inverse

Binary Operation

A binary operation, on a set A is a function which sends A × A (the set of all ordered pairs of A) back to A. In less formal terms, it takes in two of the elements from A & returns a single element which also exists in A. Another way to look at this, is that the operation in question has the property of closure.

A good example to think of, is addition of the set of integers: is addition a binary operation?

screen-shot-2017-03-04-at-3-39-54-pm

Some test cases for addition of integers

It would certainly seem to be the case! It is, & we intuitively know this to be true (which is why I am going to skip the proof).

Associativity

Say we have a binary operation on the set A. This operation is said to associative if, for all a, b, cA

(a b) ∗ c  = a ∗ (bc)

If we use our previous example about addition on the set of integers, we can easily see that addition on ℤ is associative. But what about subtraction?

screen-shot-2017-03-04-at-4-14-52-pm

Using Mathematica, we can gauge whether or not two statements are equivalent using the triple ‘=’ symbol; explicitly, the left-hand side is equal to a-b-c, while the right-hand side is equal to a-b+c

So we can confidently say that subtraction on the set of integers is not associative. So while it is a binary operation, <ℤ, -> still fails to be a group.

Identity Element

Suppose again that ∗ is a binary operation on A. An element e of A is said to be an identity for ∗, for all aA if

a ∗ e = e ∗ a = a

Using our handy example of the set of integers, ℤ, we know that <ℤ, +> does have an identity element. Similarly, <ℤ, ×> also has an identity element:

screen-shot-2017-03-04-at-4-23-54-pm

Inverses

Suppose, once more, that ∗ is a binary operation on A, with identity element e. Let aA. An element b of A is said to be an inverse of a wrt ∗ if

a ∗ b = b ∗ a = e

The set of integers does in fact have inverses under addition as well. In fact, we are very familiar with these inverses:

screen-shot-2017-03-04-at-4-29-07-pm

I will let you, the attentive reader, decide for yourself if you think that <ℤ, ×> also contains inverses. What about <ℤ, ->?

Subgroups

It might not seem like it, but subgroups are an important aspect of group theory. But why look at subgroups when you could just look at groups? After all, the term itself implies that it is not even a whole group…surely there must be more information contained in a regular ol’ group? That was a question I had asked myself, too; I just didn’t get the point.

Now though, I realize that you can learn a lot about the structure of a group by analyzing their subgroups. More generally, if you want to understand any class of mathematical structures, it helps to understand how objects in that class relate to one another. With groups, this begs the question: Can I build new groups from old ones? Subgroups help us answer this question.

Suppose G is a group. A subset H of G is called a subgroup of G if

  • H is non-empty
  • H is closed under the operation of G
  • H is closed under taking inverses

Non-empty

All this means is that there must be at least one element in H; it cannot be the empty set, ∅.

Closure

If H is a subgroup of G, we sometimes say that H inherits the operation of G.

Let’s look at this idea with some things we are familiar with. Particularly, let us use our handy dandy knowledge about the set of integers ℤ under addition. We know that a subset of ℤ could be S = {-1, 0, 1}. If we add any of the elements in S together, we will always end up with an element that is also in S.

That is pretty easy to see, so let’s look at a counter-example. Let us have a second subset of ℤ, T = {-3, -2, -1, 0, 1, 2, 3}. At first glance, it would seem like T is closed under addition on ℤ. However, if we add 2 and 3 together, that would result in 5. And we can see that 5 ∉ T.

Therefore, T does not inherit the operation of ℤ.

Inverses…Part Two

If H is a subset of G, then any element within H must have its respective inverse also be in H. We talked about what inverses were a bit earlier, so I am not going to re-type it.

You might be reading this and be thinking, “But wait! Shouldn’t a subgroup also contain the identity element as well? Silly Jesse…you buffoon…” Have no fear! For the existence of an identity element in H actually follows from the existence of inverses. I won’t prove it, but please trust me…

Special Groups and Their Properties

There are certain groups in which interesting patterns crop up. This makes them stand out amongst other groups, thus they demand special attention be paid to them. One such group is called a cyclic group.

Let G be a group with aG. The subgroup <a> is called the cyclic subgroup generated by a. If this subgroup contains all the elements of G, then G is also cyclic. You can inherently see the usefulness of subgroups in full effect here: the ability to understand more about a group can come from the existence of certain subgroups.

But what exactly does it mean for a group to be generated by an element a? In short, it means that all the elements of a group can be represented as a multiplier of a single element of that group.

More formally, ∃ m ∈ ℤ such that G = {a}, or in additive notation, G = {ma} .

Let’s take a look at this idea in the context of an example: Take the group <ℤ8, +> that is, {0, 1, 2, 3, 4, 5, 6, 7} under the operation of addition.

Screen Shot 2017-03-06 at 3.54.19 PM

Additive table of integers modulo (order) 8 & associated generators of the group. Elements {1, 3, 5, 7} can be multiplied by some m (mod 8) that result in the creation of every other element in the group.

Think about it: if we took m × 2, we would only end up with multiples of 2 – {0, 2, 4, 6} – which does not account for every element in ℤ8. Moreover, the same can be said about elements 4 and 6 – they do not generate every other element in the group.

Why are the generators the ones that they are? What is so special about the relationship between the generating elements of a group & the order of the group? Well, much to the dismay of physics students everywhere, the result is pretty interesting…

Relation to Number Theory

While there are multiple connections between algebraic structures – such as groups – to number theory, perhaps the most useful one comes from ideas related to the division of numbers. In particular, the notion of relative primality is especially useful for understanding more about behavior of cyclic groups.

Recall that m divides n (n | m) if there exists a k such that n = mk. Also recall, that an integer p is prime if p has exactly two divisors: p & 1.

Now, if both n & m are not zero, then ∃ d ∈ ℤ+ for which d is the greatest common divisor of n & m:  gcd(n, m) = d.

When d is equal to 1, we say that n & m are relatively prime. Meaning, there are no factors shared between n & m, which can divide the both of them, other than 1.

Going back to our previous example of cyclic groups, using <ℤ8, +>, what do we notice about the generators?

Screen Shot 2017-03-06 at 4.32.55 PM

The gcd between the generators of integers modulo 8 & 8 is 1.

More generally, the gcd between the generators of any group, & the order of the group will always be 1. If you did not know what the generators of a cyclic group were, you could find them using this concept.

There is more to be said about the relation between number theory & group theory – such as the use of the division algorithm to prove existence of particular elements in cyclic subgroups – but I feel like I’ve already made a compelling enough case for the utility of number theory.

Like always, thanks for reading!

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