Introducing Groups to Beginners

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1. Introduction. We will do some group theory. Here "group" refers to a "group of symmetry transformations", and we should think of elements of the group as functions mapping an object to itself in some particularly symmetric way.

2. Definition. A Group consists of a set ${G}$ equipped with

1. a law of composition ${\circ\colon G\times G\to G}$,
2. an identity element ${e\in G}$, and
3. an inverse operator ${(-)^{-1}\colon G\to G}$

such that

1. Associativity: For any ${g_{1}}$, ${g_{2}}$, ${g_{3}\in G}$, ${(g_{1}\circ g_{2})\circ g_{3}=g_{1}\circ(g_{2}\circ g_{3})}$
2. Unit law: For any ${g\in G}$, ${g\circ e=e\circ g=g}$
3. Inverse law: For any ${g\in G}$, ${g^{-1}\circ g=g\circ g^{-1}=e}$.

3. Effective Thinking Principle: Create Examples. Whenever encountering a new definition, it's useful to construct examples. Plus, it's fun. Now let us consider a bunch of examples!

4. Example (Trivial). One strategy is to find the most boring example possible. We can't use ${G=\emptyset}$ since a group must contain at least one element: the identity element ${e\in G}$. Thus the next most boring candidate is the group containing only the identity element ${G=\{e\}}$. This is the Trivial Group.

5. Example (Dihedral). Consider the regular ${n}$-gon in the plane ${X\subset\mathbb{R}^{2}}$ with vertices located at ${(\cos(k2\pi/n), \sin(k2\pi/n))}$ for ${k=0,1,\dots,n-1}$. We also require ${n\geq3}$ to form a non-degenerate polygon (${n=2}$ is just a line segment, and ${n=1}$ is one dot).

We can rotate the polygon by multiples of ${2\pi/n}$ radians. There are several ways to visualize this, I suppose we could consider rotations of the plane by ${2\pi/n}$ radians:

$$r\colon\mathbb{R}^{2}\to\mathbb{R}^{2} \tag{1}$$ which acts like the linear transformation $$r \begin{pmatrix} x\\ y \end{pmatrix} := \begin{pmatrix} \cos(2\pi/n) & -\sin(2\pi/n)\\ \sin(2\pi/n) & \cos(2\pi/n) \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}. \tag{2}$$ We see that the image of our ${n}$-gon under this transformation ${r(X)=X}$ remains invariant.

The other transformation worth exploring is reflecting about the ${x}$-axis, ${s\colon\mathbb{R}^{2}\to\mathbb{R}^{2}}$ which may be defined by

$$s \begin{pmatrix} x\\ y \end{pmatrix} := \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}. \tag{3}$$ This transformation also leaves our polygon invariant ${s(X)=X}$.

We can compose these two types of transformations. Observe that ${s\circ s=\mathrm{id}}$ and the ${n}$-fold composition ${r^{n}=r\circ\dots\circ r=\mathrm{id}}$ both yield the identity transformation ${\mathrm{id}(x)=x}$ for all ${x\in\mathbb{R}^{2}}$. Then we have ${2n}$ symmetry transformations: ${\mathrm{id}}$, ${r}$, ..., ${r^{n-1}}$; and ${s}$, ${s\circ r}$, ..., ${s\circ r^{n-1}}$. What about, say, ${r\circ s}$? We find ${s\circ r^{k}\circ s=r^{-k}}$, so ${r^{k}\circ s = s\circ r^{-k}}$. Thus it's contained in our list of symmetry transformations.

The symmetry group thus constructed is called the Dihedral Group. Geometers denote it by ${D_{n}}$, algebraists denote it by ${D_{2n}}$, and we denote it by ${D_{n}}$.

6. Example (Rotations of regular polygon). We can restrict our attention, working with the previous example further, to only rotations of the regular ${n}$-gon by multiples of ${2\pi/n}$ radians. We can describe this group as "generated by a single element", i.e., symmetries are of the form ${r^{k}}$ for ${k\in\mathbb{Z}}$. This is an example of a Cyclic Group. In particular, it is commutative: any symmetries ${r_{1}}$ and ${r_{2}}$ satisfy ${r_{1}\circ r_{2}=r_{2}\circ r_{1}}$. These are special situations, let us carve out space to define these concepts explicitly.

7. Definition. We call a group ${G}$ Abelian if it is commutative, i.e., for any transformations ${f}$, ${g\in G}$ we have ${f\circ g = g\circ f}$. In this case, we write ${f\circ g}$ as ${f+g}$, using the plus sign to stress commutativity.

8. Definition. We call a group ${G}$ Cyclic if there is at least one element ${g\in G}$ such that ${\{g^{n}\mid n\in\mathbb{Z}\}=G}$ the entire group consists of iterates of ${g}$ and ${g^{-1}}$.

9. Example (Number Systems). Another few examples the reader may know are the familiar number systems under addition: the integers ${\mathbb{Z}}$, the rational numbers ${\mathbb{Q}}$, the real numbers ${\mathbb{R}}$, and the complex numbers ${\mathbb{C}}$. They are commutative groups.

10. Example (Infinite dihedral). We can take the infinite limit of the dihedral group to get the infinite dihedral group ${D_{\infty}}$. We formally describe it as consisting of "rotations" ${r}$ and "reflections" ${s}$ such that

1. ${r^{m}\circ r^{n} = r^{m+n}}$ for any ${m}$, ${n\in\mathbb{Z}}$;
2. ${s\circ r^{m}\circ s = r^{-m}}$ for any ${m\in\mathbb{Z}}$;
3. ${s\circ s = e}$;
4. ${r^{n}\circ r^{-n} = r^{-n}\circ r^{n} = e}$ for any ${n\in\mathbb{Z}}$, in particular ${r^{0}=e}$.

In this sense, the "infinite limit" turns rotations into something like the integers.

11. Example (Circular dihedral). A more intuitive "infinite limit" of the dihedral group is the symmetries of the unit circle ${S^{1}}$ in the plane ${\mathbb{R}^{2}}$. These are anti-clockwise rotations and reflection about the ${x}$-axis, but rotations are parametrized by a real parameter (the "angle"):

$$r_{\theta}~``=\!\!\mbox{"} \begin{pmatrix}\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta) \end{pmatrix}. \tag{4}$$ Here we write an "equals" sign in quotes because this is the intuition. A group is abstract, whereas the matrix is a concrete realization of the symmetry.

The reader should verify the axioms for a group are satisfied, with the hint that ${r_{\theta}\circ r_{\phi} = r_{\theta+\phi}}$ and the usual relation between reflection and rotation holds.

This group is called the Orthogonal Group in 2-dimensions.

Exercises

Exercise 1. Is ${\mathbb{Z}}$ a cyclic group? Is ${\mathbb{C}}$ a cyclic group?

Exercise 2. Is the non-negative integers ${\mathbb{N}_{0}}$ a group under addition? Under multiplication?

Exercise 3. Are the positive real numbers ${\mathbb{R}_{\text{pos}}}$ a group under multiplication?

Exercise 4. Pick your favorite polyhedron in 3-dimensions. Determine its symmetry group.

Exercise 5. Complex conjugation acts on ${\mathbb{C}}$ by sending ${x+i\cdot y}$ to ${x-i\cdot y}$. Does this give us a symmetry group?

Exercise 6 (challenging). If we consider polynomials with coefficients in, say, rational numbers (denoted ${\mathbb{Q}[x]}$ for polynomials with the unknown ${x}$), then how can we form a symmetry group of ${\mathbb{Q}[x]}$?

Exercise 7 (General Linear Group). Take ${n\in\mathbb{N}}$ to be a fixed positive integer, preferably ${n\geq2}$. Consider the collection of invertible ${n}$-by-${n}$ matrices with entries which are rational numbers

$${\mathrm{GL}(n, \mathbb{Q}) = \{ M\in\mathrm{Mat}(n\times n, \mathbb{Q}) \mid \det(M)\neq0\}.}$$ Prove this is a group under matrix multiplication.