Group Theory

1. Symmetries of an Object
2. Notes to Self

1. Symmetries of an Object

Suppose we have an object (like a regular polygon in a plane). What is a "symmetry" of our object?

1.1. Necessary Qualities of a Symmetry

Presumably it is some kind of transformation. For this to make sense, we extend the transformation to the plane. A symmetry should "leave the object invariant". What does that mean?

It is necessary to say the transformation \(f\colon\RR^{2}\to\RR^{2}\) leaves our polygon \(P\subset\RR^{2}\) invariant if \(f(P)=P\). So this is equivalent to saying \(f\) is surjective. But is this sufficient?

Well, if \(f\) is surjective, it has a right-inverse \(h\colon\RR^{2}\to\RR^{2}\) (which satisfies \((f\circ h)(p)\) for every \(p\in P\)). We want this right-inverse to also be a symmetry. Which implies \(h\) has a right-inverse, call it \(g\colon\RR^{2}\to\RR^{2}\). Then what is \(f\circ h\circ g\)? Well, we find \((f\circ(h\circ g))(x)=f(x)\) but also \(((f\circ h)\circ g)(x) = g(x)\) imply \(f=g\). Hence every symmetry is an invertible function.

1.2. A Collection of Symmetries

Now, we have described what qualifies as a symmetry of an object, suppose we have collected all the symmetries for an object. What happens if we compose two symmetries together? I.e., if \(f_{1},f_{2}\colon P\to P\) are two symmetries, what about \(f_{2}\circ f_{1}\)? Is that also a symmetry or not?

Well, its inverse is the composition of inverses in reverse order. If our collection of symmetries includes the inverse symmetries, then the set of symmetries is closed under function composition.

This gives us grounds for generalizing these qualities into an abstract notion of a group.

A Group consists of a set \(G\) equipped with

a binary operator \(\mu\colon G\times G\to G\),
an inverse operator \(\iota\colon G\to G\), and
an identity element \(e\in G\)

such that

Multiplication is associative: \(\mu\circ(\mathrm{id}\times\mu) = \mu\circ(\mu\times\mathrm{id})\) so \((\mu\circ(\mathrm{id}\times\mu))(x,y,z) = \mu(x, \mu(y,z))\) is equal to \((\mu\circ(\mu\times\mathrm{id}))(x,y,z)=\mu(\mu(x,y),z)\)
The identity element \(\mu(e,x)=\mu(x,e)=x\)
The inverse map: \(\mu(\iota(x),x)=\mu(x,\iota(x))=x\).

This mystifies the motivation of studying groups under the shroud of abstraction, but we can think about groups in general settings.

1.3. Example: Symmetries of Polynomials

Thus far we have intuitively pictured symmetries of geometric objects. What about symmetries of nongeometric object? Like…polynomials.

The first question we should ask ourselves is: what is a polynomial? Like, how do we represent a polynomial? Because our symmetries would act on this "encoding".

We could use the fundamental theorem of algebra to write a polynomial as

\begin{equation} p(x) = p_{n}x^{n} + \dots + p_{1}x + p_{0} = c(x - r_{1})(\dots)(x - r_{n}) \end{equation}

where \(r_{k}\in\mathbb{C}\) are the roots of the polynomial. One obvious symmetry amounts to permuting the roots of the polynomial.

2. Notes to Self

If we take our abstract definition of a group, we can define a "group object" internal to any sufficiently nice category. This gives us the various different sorts of groups (e.g., a group object internal to the category of topological spaces is a topological group, a group object internal to the category of manifolds is a [real] Lie group, etc.).

The group object internal to Grp the category of groups is itself an Abelian group. This is a corollary of the Eckmann–Hilton argument.