Group Morphisms

1. There is some "Kabuki theater" whenever introducing a new mathematical gadget, thanks to category theory: we have our new gadget, then we could ask about morphisms (very important), "subgadgets", and universal constructions (products, quotients, etc.). The exciting thing, as in Kabuki theater, is the order and manner of presentation.

2. Definition. Let ${G}$ and ${H}$ be groups. We define a Group Morphism to consist of a function ${\varphi\colon G\to H}$ of the underlying sets such that

1. group operation is preserved: for any ${g_{1}}$, ${g_{2}\in G}$, we have ${\varphi(g_{1}g_{2})=\varphi(g_{1})\varphi(g_{2})}$;
2. identity element is preserved: if ${e_{G}\in G}$ is the identity element of ${G}$ and ${e_{H}\in H}$ is the identity element of ${H}$, then ${\varphi(e_{G})=e_{H}}$;
3. inverse is preserved: for any ${g\in G}$, ${\varphi(g^{-1})=\varphi(g)^{-1}}$.

2.1. Remark. Older texts refer to group morphisms as "group homomorphisms". After category theory became popular and part of the standard curriculum, the "homo-" prefix was dropped because group morphisms live in the same category, so it was redundant.

3. Example. One "low hanging fruit" for morphism examples is the identity morphism. We should check, for any group ${G}$, the identity function ${\mathrm{id}\colon G\to G}$ is a bona fide group morphism.

We see it preserves the group operation. For any ${g_{1}}$, ${g_{2}\in G}$, we have ${\mathrm{id}(g_{1}g_{2})=g_{1}g_{2}}$ by definition of the identity function. But this is also equal to ${\mathrm{id}(g_{1})\mathrm{id}(g_{2})}$. Thus the group operation is preserved.

The identity element is preserved ${\mathrm{id}(e_{G})=e_{G}}$.

Inversion is also preserved ${\mathrm{id}(g^{-1})=g^{-1}=\mathrm{id}(g)^{-1}}$ for any ${g\in G}$.

Thus taken together, it follows the identity mapping satisfies the axioms of a group morphism.

4. Example. Let ${G}$ be any group, and consider ${\mathbb{Z}}$ equipped with addition as a group. For each ${g\in G}$, we have a group morphism ${\varphi\colon\mathbb{Z}\to G}$ sending ${1\in\mathbb{Z}}$ to ${g\in G}$. Is this really a group morphism?

We can check that the properties are (or, ought to be) satisfied. If the group operation is preserved, then ${\varphi(1+1)=\varphi(1)\varphi(1)=g^{2}}$ and more generally, for any ${m\in\mathbb{Z}}$, we have ${\varphi(m+1)=\varphi(1)^{m}=g^{m}}$.

For the identity element being preserved, that means ${\varphi(0)=e_{G}}$, which is fine: it corresponds to ${g^{0}=e_{G}}$.

Group inverses would be ${\varphi(-m)=\varphi(m)^{-1}=(g^{m})^{-1}}$. And we know this is precisely the same as ${g^{-m}}$.

5. Example. Consider the group ${\mathrm{GL}(2,\mathbb{R})}$ and the multiplicative group ${\mathbb{R}^{\times}}$ of nonzero real numbers. Then the determinant $$ \det\colon\mathrm{GL}(2,\mathbb{R})\to\mathbb{R}^{\times} \tag{1}$$ is a group morphism. Let us prove it!

We see, for any matrices ${M}$, ${N\in\mathrm{GL}(2,\mathbb{R})}$ we have $$ \det(MN)=\det(M)\det(N). \tag{2}$$ This is a familiar fact in linear algebra. But for us, it tells us the group operation is preserved.

The identity element must be mapped to the identity element. We see the identity matrix ${I\in\mathrm{GL}(2,\mathbb{R})}$ has ${\det(I)=1}$. Thus the determinant preserves the group identity element.

As far as the group inverse, well, this follows from previous results, right? After all, if ${M\in\mathrm{GL}(2,\mathbb{R})}$, then ${M^{-1}\in\mathrm{GL}(2,\mathbb{R})}$, and $$ I = M^{-1}M \tag{3}$$ so $$ \det(M^{-1}M)=\det(M^{-1})\det(M)=1 \tag{4}$$ and thus by division $$ \det(M^{-1})=\det(M)^{-1}. \tag{5}$$ Thus the group inverse operator is preserved.

6. Definition. Let ${\varphi\colon G\to H}$ be a group morphism. We define the Kernel of ${\varphi}$ to be the pre-image of the identity element of ${H}$: \begin{equation*} \ker(\varphi)=\{g\in G|\varphi(g)=e_{H}\}. \end{equation*}

7. Example. For the group morphism ${\det\colon\mathrm{GL}(2,\mathbb{R})\to\mathbb{R}^{\times}}$, the kernel would be $$ \ker(\det)=\{M\in\mathrm{GL}(2,\mathbb{R})|\det(M)=1\}. \tag{6}$$ That is to say, it consists of matrices with unit determinant. Observe, this is a group under matrix multiplication: if two matrices have unit determinant, their product has unit determinant; the identity matrix is in the kernel; and it's closed under inverses. This is an important group called the Special Linear Group, denoted ${\mathrm{SL}(2,\mathbb{R})}$.

1. Properties of Morphisms

8. Proposition. Let ${\varphi\colon G\to H}$ be a group morphism. If ${g\in G}$ is any element, then ${\varphi(g^{-1})=\varphi(g)^{-1}}$.

Proof: Let ${g\in G}$ (so ${\varphi(g)\in H}$). We find ${\varphi(g\cdot g^{-1})=\varphi(g)\varphi(g^{-1})=e_{H}}$, thus multiplying on the left by ${\varphi(g)^{-1}}$ gives the result. ∎

9. Proposition. Let ${\varphi\colon G\to H}$ be a group morphism. If ${g\in G}$ is any element and ${n\in\mathbb{Z}}$ is any integer, then ${\varphi(g^{n})=\varphi(g)^{n}}$.

Proof: Per cases since ${n<0}$ or ${n=0}$ or ${n>0}$. The ${n=0}$ case is obvious.

For ${n>0}$, by induction. The base case ${n=1}$ gives ${\varphi(g^{1})=\varphi(g)^{1}}$, which is obvious. Assume this holds for arbitrary ${n}$. Then the inductive case ${n+1}$ is $$ \varphi(g^{n+1})=\varphi(g^{n}g)=\varphi(g)^{n}\varphi(g)=\varphi(g)^{n+1}. \tag{7}$$ Thus we have proven the result for non-negative ${n}$.

For negative ${n\in\mathbb{Z}}$, the proof is analogous. ∎

10. Theorem. The composition of group morphisms is a group morphism. More explicitly, if ${\varphi\colon G\to H}$ and ${\psi\colon H\to K}$ are group morphisms, then ${\psi\circ\varphi\colon G\to K}$ is a group morphism.

11. Theorem. Let ${\varphi\colon G\to H}$ be a group morphism. If ${\ker(\varphi)=\{e_{G}\}}$, then ${\varphi}$ is injective.

Proof: Assume ${\ker(\varphi)=\{e_{G}\}}$. Let ${g_{1}}$, ${g_{2}\in G}$ be completely arbitrary. (We want to show if ${\varphi(g_{1})=\varphi(g_{2})}$, then ${g_{1}=g_{2}}$.) Assume ${\varphi(g_{1})=\varphi(g_{2})}$. Then ${\varphi(g_{1})\varphi(g_{2})^{-1}=e_{H}}$ by multiplying both sides on the right by ${\varphi(g_{2})^{-1}}$. And ${\varphi(g_{1})\varphi(g_{2}^{-1})=\varphi(g_{1}\cdot g_{2}^{-1})=e_{H}}$. Thus ${g_{1}\cdot g_{2}^{-1}\in\ker(\varphi)}$ by definition of the kernel. But we assumed the only member of the kernel was identity element. Thus ${g_{1}\cdot g_{2}^{-1}=e_{G}}$, and moreover ${g_{1}=g_{2}}$. Hence ${\varphi}$ is injective. ∎

2. Exercises

Exercise 1. Let ${G}$ be a group, ${n\in\mathbb{N}}$ be a fixed positive integer. Prove or find a counter-example: ${\varphi\colon G\to G}$, sending ${g}$ to ${\varphi(g)=g^{n}}$ is a group morphism.

Exercise 2. Prove or find a counter-example: the matrix trace ${\mathop{\rm tr}\nolimits\colon\mathrm{GL}(2,\mathbb{R})\to\mathbb{R}}$ is a group morphism.

Exercise 3. Is the exponential function on the real numbers a group morphism ${\exp\colon\mathbb{R}\to\mathbb{R}^{\times}}$?

Exercise 4. Is the matrix exponential a group morphism ${\exp\colon\mathrm{Mat}_{2}(\mathbb{R})\to\mathrm{GL}(2,\mathbb{R})}$? [Hint: $\mathrm{Mat}_{2}(\mathbb{R})$ has its group operation be matrix addition. Is this preserved?]

Typography of Lie groups, Lie algebras

This is a note for myself as I review my notes on finite groups, Lie groups (both the classic infinite ones and the finite simple groups of Lie type), Lie algebras, and specifically the typography for them.

First, no one agrees completely, and everyone's conventions is slightly different. What I mean by this: although everyone agrees, e.g., that the A_n family of Lie algebras refers to the same thing, some people use serif font for the "A", others use bold, some italicize, others do not, etc. Each field uses its own notation with good reason, but I think we can standardize notation a bit better in group theory.

I'm inclined to follow Robert Wilson's conventions from his book The Finite Simple Groups (2009).

Families of Groups

Families of Simple Lie groups: use upright, teletype font for the family and "math italics" for the subscript if it's a variable, e.g., $\mathtt{A}_{n}$, $\mathtt{B}_{5}$, $\mathtt{C}_{m}$, $\mathtt{D}_{n^{2}}$, $\mathtt{E}_{8}$

Exceptional Finite Simple Groups of Lie Type: don't treat the formatting as special, so, e.g., the Steinberg groups would be ${}^{2}A_{n}(q^{2})$, ${}^{2}D_{n}(q^{2})$, ${}^{2}E_{6}(q^{2})$, ${}^{3}D_{4}(q^{3})$.

Sporadic simple group: these should be made upright, e.g., the Suzuki group is $\mathrm{Suz}$, the Matthieu groups look like $\mathrm{M}_{11}$, the Conway groups $\mathrm{Co}_{1}$ and $\mathrm{Co}_{2}$, and so on. BUT the exception to this rule is that the Monster group is written $\mathbb{M}$ and the Baby Monster $\mathbb{B}$.

Alternating, Cyclic, Symmetric group. These are just written as $A_{n}$, $C_{n}$, or $S_{n}$. The dihedral group, too, is $D_{n}$.

Classical Lie Groups: Here there is a double standard. For classical Lie groups over the reals or complex numbers, we write something of the form $\mathrm{GL}(n, \mathbb{F})$, $\mathrm{SL}(n, \mathbb{F})$, $\mathrm{U}(n, \mathbb{F})$, $\mathrm{SU}(n, \mathbb{F})$, $\mathrm{O}(n, \mathbb{F})$, $\mathrm{SO}(n, \mathbb{F})$, $\mathrm{Sp}(n)=\mathrm{USp}(n)$ for the compact Symplectic group, $\mathrm{Sp}(2n,\mathbb{F})$ for the generic Symplectic group.

The finite groups corresponding to these guys are written a little differently in my notes: the $n$ parameter is pulled out as a subscript, because frequently we write $q$ instead of $\mathbb{F}_{q}$ for finite fields...and then looking at $\mathrm{SL}(8,9)$ is far more confusing than $\mathrm{SL}_{8}(9)$. Thus we have $\mathrm{GL}_{n}(q)$, and so on.

Projective classical groups: the projective classical groups are prefixed by a "P", not a blackboard bold $\mathbb{P}$. E.g., $\mathrm{PSL}_{2}(7)$. At present, the projective orthogonal group wikipedia page seems to agree with this convention.

Operations

For finite groups: the Atlas of finite groups seems to have set the standard conventions for finite groups, Wilson changes them slightly. We'll find $G = N{:}H$ for the semidirect product $G = N \rtimes H = H\ltimes N$. Also $A\mathop{{}^{\textstyle .}}\nolimits B = A{\,}^{\textstyle .} B$ for a non-split extension with quotient $B$ and normal subgroup $A$, but no subgroup $B$. And $A{.}B$ is an unspecified extension.

Just a quick remark concerning $A^{\textstyle{.}}B$, I was unsatisfied with this, and asked TeX.stackexchange for help. The answers were fantastic! I'll quote the two solutions submitted:

% solution one
\makeatletter
\newcommand*{\udot}{{\mathpalette\ud@t\relax}} % or \mathrel/\mathbin/etc?
\newcommand*{\ud@t}[2]{%
   \sbox\z@{\m@th$#1.$}%
   \sbox\tw@{$#1:$}%
   \raise\dimexpr\ht\tw@-\ht\z@\relax\box\z@
}
% Example usage:
%
% $A . A {:} A \udot A$\par
% $\scriptstyle A . A {:} A \udot A$\par
% $\scriptscriptstyle A . A {:} A \udot A$
\makeatother

% solution two
\makeatletter
\newcommand{\udot}{\mathpalette\udot@\relax}
\newcommand{\udot@}[2]{%
  \begingroup
  \sbox\z@{$#1{:}$}%
  \sbox\tw@{$#1{.}$}%
  \raisebox{\dimexpr\ht\z@-\ht\tw@}{$\m@th#1.$}%
  \endgroup
}
\makeatother
\newcommand{\genericextension}[1]{%
  \mathbin{%
    \nonscript\mkern-0.75\medmuskip
    {#1}%
    \nonscript\mkern-0.75\medmuskip
  }%
}
\DeclareRobustCommand{\gext}{\genericextension{.}}
\DeclareRobustCommand{\sdp}{\genericextension{:}}
\DeclareRobustCommand{\nsext}{\genericextension{\udot}}
% Example usage:
%
% $A\gext B$ is a group extension
%
% $A\sdp B$ is a semidirect product
% 
% $A\nsext B$ is a nonsplit group extension
%
% ${:}{\udot}$ for checking the alignment
% 
% $X_{A\gext B} \quad X_{A\sdp B} \quad X_{A\nsext B}$

Lie algebras. Writing notes on paper, for a given Lie group $G$, I write $\mathrm{Lie}(G)$ as its Lie algebra. (It turns out to be a functor...neat!) If I have to write Fraktur by hand, I approximate it using Pappus's caligraphy tutorial.

Charts and Atlases, Manifolds and Smooth Structures

1. Manifolds

We will introduce the machinery necessary for defining a smooth manifold.

1.1. Charts

1. Definition. Let ${X\subset M}$ be some set. An ${n}$-dimensional Chart consists of

1. an open subset ${U\subset\mathbb{R}^{n}}$
2. a map ${\varphi\colon U\to X}$

such that ${\varphi}$ is an appropriate isomorphism (for topological manifolds, it is a homeomorphism; smooth manifolds require a diffeomorphism; and so on).

2. Remark. We call ${\varphi\colon U\to X}$ a Parametrization of ${X}$, and ${\varphi^{-1}\colon X\to U}$ a Local System of Coordinates.

3. Remark. Since ${\varphi}$ is an isomorphism, the literature mixes up using ${U\to X}$ and ${X\to U}$. Milner uses ${\varphi\colon U\to X}$, but John Lee uses the opposite convention.

4. Definition. Let ${(U,\varphi)}$, ${(V,\psi)}$ be two charts. We say they are Compatible if

1. the set ${(\varphi^{-1}\circ\psi)(V)\subset U}$ is an open set;
2. the set ${(\psi^{-1}\circ\varphi)(U)\subset V}$ is an open set;
3. the map ${\psi^{-1}\circ\varphi\colon\varphi^{-1}(\psi(V))\to\psi^{-1}(\varphi(U))}$ is smooth; and
4. the map ${\varphi^{-1}\circ\psi\colon\psi^{-1}(\varphi(U))\to\varphi^{-1}(\psi(V))}$ is smooth.

In particular, the charts are compatible if ${\varphi(U)\cap\psi(V)=\emptyset}$ is disjoint.

5. Remark. We refer to the maps ${\psi^{-1}\circ\varphi}$ as Transition Functions. The condition of smooth is ${C^{\infty}(\mathbb{R}^{n})}$, but different manifolds have different conditions (we could have ${C^{k}}$ charts, or ${C^{0}}$ charts, or analytic ${C^{\omega}}$ charts, or...).

In the older literature (e.g., Kobayashi and Nomizu's Foundations of Differential Geometry), the collection of transition functions form a gadget called a Pseudogroup.

6. Remark. We abuse notation, and could be more explicit by writing $$ \psi^{-1}\circ\varphi\colon\varphi^{-1}(\varphi(U)\cap\psi(V))\to\psi^{-1}(\varphi(U)\cap\psi(V)) \tag{1}$$

▸ Exercise 1. Prove chart compatibility is an equivalence relation.

1.2. Atlases

7. Definition. Let ${M}$ be a set. An (${n}$-dimensional) Atlas consists of a collection ${\{(U_{\alpha},\varphi_{\alpha})\mid\alpha\in A\}}$ of ${n}$-dimensional charts on ${M}$ such that

1. Covers ${M}$: $\displaystyle{\bigcup_{\alpha\in A}\varphi_{\alpha}(U_{\alpha})=M}$
2. Pairwise compatible: for any ${\alpha}$, ${\beta\in A}$ the charts ${(U_{\alpha},\varphi_{\alpha})}$ and ${(U_{\beta},\varphi_{\beta})}$ are compatible.

8. Definition. Two ${n}$-dimensional atlases on ${M}$, ${\mathcal{A}}$ and ${\mathcal{B}}$, are called Equivalent if their union ${\mathcal{A}\cup\mathcal{B}}$ is also an atlas. That is to say, if any chart of ${\mathcal{A}}$ is compatible with any chart of ${\mathcal{B}}$.

9. Remark. Remember: charts are compatible, but atlases are equivalent.

10. Lemma. Let ${\mathcal{B}}$ be an atlas, let ${(U,\varphi)}$ and ${(V,\psi)}$ be two charts not contained in ${\mathcal{B}}$. If ${(U,\varphi)}$ is compatible with every chart of ${\mathcal{B}}$, and if ${(V,\psi)}$ is compatible with every chart of ${\mathcal{B}}$, then ${(U,\varphi)}$ is compatible with ${(V,\psi)}$.

11. Theorem. Equivalence of atlases is an equivalence relation.

Proof: Let ${\mathcal{A}}$, ${\mathcal{B}}$, ${\mathcal{C}}$ be arbitrary atlases on ${M}$.

1. Reflexivity: ${\mathcal{A}}$ is equivalent to itself, since by definition any pair of charts in ${\mathcal{A}}$ are compatible.
2. Symmetry: let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, then ${\mathcal{B}}$ and ${\mathcal{A}}$ are equivalent atlases.
3. Transitivity: this is the nontrivial part. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, and ${\mathcal{B}}$ be equivalent to ${\mathcal{C}}$. Then transitivity follows by considering arbitrary charts ${(U,\varphi)\in\mathcal{A}}$ and ${(V,\psi)\in\mathcal{C}}$, then applying Lemma 10.

Thus "equivalence of atlases" forms an equivalence relation. ∎

12. Proposition. The collection of atlases on a given set ${M}$ is a set, not a proper class.

Proof: The class of atlases is a subcollection of $$ \mathcal{X}=\mathcal{P}\left(\bigcup_{U\in\mathcal{P}(\mathbb{R}^{n})}\mathop{\rm Hom}\nolimits(U,M)\right) \tag{2}$$ where ${\mathop{\rm Hom}\nolimits(U,\mathbb{R}^{n})}$ is the collection of (appropriately smooth, or continuous, or holomorphic, or...) functions from ${U}$ to ${M}$. By ZF axioms, ${\mathcal{X}}$ is a set. ∎

1.3. Manifolds

13. Definition. Let ${M}$ be a set, let ${\mathcal{A}}$ be an ${n}$-dimensional atlas on ${M}$. We call a subset ${B\subset M}$ Open (with respect to ${\mathcal{A}}$) if for any chart ${(U,\varphi)\in\mathcal{A}}$ the preimage ${\varphi^{-1}(B)}$ is open (in ${U}$, and thus open in ${\mathbb{R}^{n}}$). In particular, the images ${\varphi(U)}$ are open.

14. Theorem. If two atlases ${\mathcal{A}_{1}}$ and ${\mathcal{A}_{2}}$ on ${M}$ are equivalent, then a subset ${B\subset M}$ is open with respect to ${\mathcal{A}_{1}}$ if and only it is open with respect ${\mathcal{A}_{2}}$.

15. Remark. This theorem shows an equivalence class of atlases on ${M}$ makes ${M}$ a topological space. We may therefore meaningfully speak about topological properties of ${M}$ (like compactness, connectedness, and so forth).

16. Corollary. Let ${\mathcal{A}}$ be an ${n}$-dimensional atlas for ${M}$. Then the collection of open sets with respect to ${\mathcal{A}}$ form a topology on ${M}$.

17. Definition. Let ${M}$ be a fixed set. A ${n}$-Dimensional Differential Structure (or ${n}$-Dimensional Smooth Structure) on ${M}$ consists of an equivalence class ${\mathfrak{D}}$ of ${n}$-dimensional atlases on ${M}$ such that

1. Second-Countable: ${\mathfrak{D}}$ contains an at most countable atlas;
2. Hausdorff: for any distinct ${p,q\in M}$, there exists disjoint open neighborhoods ${U,V\subset M}$ such that ${p\in U}$ and ${q\in V}$.

18. Remark (Smooth Structure using a Maximal Atlas). Equivalence classes are awkward to work with, and so it is more popular to consider maximal atlases. An atlas ${\mathcal{A}}$ is maximal if it contains all charts compatible with every chart in ${\mathcal{A}}$. Given an equivalence class ${\mathfrak{A}}$ of atlases, we may obtain a maximal atlas by considering $$ \mathcal{A}_{\text{max}} = \bigcup_{\mathcal{A}\in\mathfrak{A}}\mathcal{A}. \tag{3}$$ This may be used instead of an equivalence class of atlases in defining a differential structure, provided the second-countable axiom is reworded as: ${\mathcal{A}_{\text{max}}}$ contains an at most countable subatlas.

19. Remark (Convenient Fiction). No one actually constructs either a maximal atlas or a differential structure. We typically construct a smooth atlas on ${M}$, then announce we are working with the differential structure containing our atlas. Thus maximal atlases and, to some degree, differential structures are a convenient fiction.

20. Definition. A (Smooth) ${n}$-Dimensional Manifold consists of a set ${M}$ equipped with an ${n}$-dimensional differential structure.

21. Puzzle. Is this definition correct? By this, I mean: is an "${n}$-Dimensional Differential Structure" actually structure (in the sense of "stuff, structure, and properties")?

Introducing Groups to Beginners

[This is an experiment to see if some software to translate LaTeX to html works.]

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.

TeX macro for normal operator ordering

I've always been bothered with normal operator ordering, writing $:O(a)O(b):$ always produces bad results.

The quick fix I've been using is the following:

\def\normOrd#1{\mathop{:}\nolimits\!#1\!\mathop{:}\nolimits}

%%
% example:
% \begin{equation}
% \normOrd{a(z)b(\omega)} = a(z)_{+}b(\omega)+(-1)^{\alpha\beta}b(\omega)a(z)_{-}
% \end{equation}
%%

Which in practice looks like:

How I got this solution

I determined this solution iteratively after many different attempts, which I shall enumerate along with the problems they each had.

However, using mere colons :a(z)b(\omega): = ... produces the following:

Being clever, I asked myself "Hey, why not write :x\colon for the normal ordering?" This was clever, but wrong. Consider the following example:

g = :x\colon

Producing:

Not one to give up easily, I found a \cocolon definition on tex.stackexchange. Trying that instead:

g = \cocolon x\colon = y

Produces strange extra whitespace on the right:

After examining the co-colon code, I just determined that something along the lines of

% rough draft definition #1
\def\normOrd#1{\mathrel{:}\!#1\!\mathrel{:}}

would work. This didn't quite work, the whitespacing was strange. So instead I just use \mathop{:}\nolimits..., which produces the desired result.