# Group Theory 5: Subgroups

In this lecture, we study subgroups. A nonempty subset $H$ of a group $G$ is called a subgroup of $G$ if $H$ itself forms a group relative to the operation $\cdot$ in $G$. If $H$ is a subgroup of $G$, we simply write $H\leq G$. Clearly $G\leq G$ and $\{e\}\leq G$. $G$ and $\{e\}$ are called trivial subgroups of $G$. Before we discuss subgroups further we need to go over some basic properties of a group as we need them.

Lemma. If $G$ is a group, then:

(a) Its identity element is unique.

(b) Each $a\in G$ has a unique inverse $a^{-1}\in G$.

(c) $\forall a\in G$, $(a^{-1})^{-1}=a$.

(d) $\forall a,b\in G$, $(ab)^{-1}=b^{-1}a^{-1}$.

The proof of this lemma is pretty much straightforward and is left to readers.

In order to show that a nonempty subset $H$ of a group $G$ is a subgroup we need to check:

1. $H$ is closed under $\cdot$, the operation in $G$. That is, $\forall a,b\in H$, $ab\in H$.

2. $e\in H$.

3. $\forall a\in H$, $a^{-1}\in H$.

Note that the associative law holds automatically. It turns out that there is a simpler criterion to check if a nonempty subset is a subgroup of a group.

Lemma. A nonempty subset $H\subset G$ is a subgroup of $G$ if and only of $\forall a,b\in H$, $ab^{-1}\in H$.

Proof. ($\Longrightarrow$) Clear.

($\Longleftarrow$) Let $a,b\in H$. Then by assumption, $aa^{-1}=e\in H$. Since $a,e\in H$, $ea^{-1}=a^{-1}\in H$. Since $b^{-1}\in H$, $a(b^{-1})^{-1}=ab\in H$. Therefore, $H$ is a subgroup of $G$.

Example. $\forall n\in\mathbb{N}\cup\{0\}$, $n\mathbb{Z}\leq(\mathbb{Z},+)$. Here, $n\mathbb{Z}=\{nx: x\in\mathbb{Z}\}$. In fact, they are all the subgroups of $(\mathbb{Z},+)$.

Definition. The cyclic subgroup of $G$ generated by $a\in G$ is $\{a^k: k\in\mathbb{Z}\}$. It is denoted by $\langle a\rangle$.

Example. Let $G$ be a group and $a\in G$. Let $C(a)=\{g\in G: ga=ag\}$. Then $C(a)\leq G$. $C(a)$ is called the centralizer of $a$ in $G$.

Example. Let $G$ be a group and let $Z(G)=\{z\in G: zx=xz\ \forall x\in G\}$. Then $Z(G)$ is a subgroup of $G$, called the center of $G$.

Let $G$ be a group and $H\leq G$. $\forall a\in G$, $a^{-1}Ha=\{a^{-1}ha: h\in H\}$ is also a subgroup of $G$.

Lemma. Suppose that $(G,\cdot)$ is a group and $H$ is a finite nonempty subset of $H$. If $H$ is closed under $\cdot$, then $H$ is a subgroup of $G$.

Proof. Let $H=\{a_1,a_2,\cdots,a_n\}$. Then $\forall a\in H$, $aH=\{aa_1,aa_2,\cdots,aa_n\}\subset H$. On the other hand, if $aa_i=aa_j$ then $a_i=a_j$. So, $|aH|=|H|$. This implies that $aH=H$ and hence
\begin{align*}
a\in aH &\Longrightarrow a=aa_i\ \mbox{for some}\ i\\
&\Longrightarrow a_i=e\in H.
\end{align*}
Since $e\in aH$, $e=aa_j$ for some $j$ $\Longrightarrow$ $a_j=a^{-1}$.

Example. The symmetric group $S_3=\{1,(123),(132),(23),(13),(12)\}$ has 5 subgroups
$\{1\}$, $\{1,(12)\}$, $\{1,(13)\}$, $\{1,(23)\}$, and $A_3=\{1,(123),(132)\}$. $A_3$ is called alternating group of degree 3. Cycles of length 2 such as (12), (13), (23) are called transpositions. Any permutation can be written as a product of either an even number of transpositions or an odd number of transpositions. If a permutation is a product of an even number of transpositions, the permutation is called an even permutation. If a permutation is a product of an odd number of transpositions, the permutation is called an odd permutation. The identity permutation 1 is an even permutation as $\begin{pmatrix} 1 & 2 & 3\\ 1 & 2 & 3 \end{pmatrix}$ can be written as (12)(21)(23)(32)(31)(13). The permutation (123) can be written as (12)(13). So, (123) is an even permutation. Similarly (132) is an even permutation as well. So it turns out that the alternating group $A_3$ is the set of all even permutations. In general, the set $A_n$ of all even permutations of $\{1,2,3,\cdots,n\}$ form a subgroup of the symmetric group $S_n$. Remember that $S_n$ has order $n!$. Since there are exactly the same number of even permutations and odd permutations, the alternating group $A_n$ of degree $n$ has order $\frac{n!}{2}$.

In this lecture, we study subgroups. A nonempty subset $H$ of a group $G$ is called a subgroup of $G$ if $H$ itself forms a group relative to the operation $\cdot$ in $G$. If $H$ is a subgroup of $G$, we simply write $H\leq G$. Clearly $G\leq G$ and $\{e\}\leq G$. $G$ and $\{e\}$ are called trivial subgroups of $G$. Before we discuss subgroups further we need to go over some basic properties of a group as we need them.

Lemma. If $G$ is a group, then:

(a) Its identity element is unique.

(b) Each $a\in G$ has a unique inverse $a^{-1}\in G$.

(c) $\forall a\in G$, $(a^{-1})^{-1}=a$.

(d) $\forall a,b\in G$, $(ab)^{-1}=b^{-1}a^{-1}$.

The proof of this lemma is pretty much straightforward and is left to readers.

In order to show that a nonempty subset $H$ of a group $G$ is a subgroup we need to check:

1. $H$ is closed under $\cdot$, the operation in $G$. That is, $\forall a,b\in H$, $ab\in H$.

2. $e\in H$.

3. $\forall a\in H$, $a^{-1}\in H$.

Note that the associative law holds automatically. It turns out that there is a simpler criterion to check if a nonempty subset is a subgroup of a group.

Lemma. A nonempty subset $H\subset G$ is a subgroup of $G$ if and only of $\forall a,b\in H$, $ab^{-1}\in H$.

Proof. ($\Longrightarrow$) Clear.

($\Longleftarrow$) Let $a,b\in H$. Then by assumption, $aa^{-1}=e\in H$. Since $a,e\in H$, $ea^{-1}=a^{-1}\in H$. Since $b^{-1}\in H$, $a(b^{-1})^{-1}=ab\in H$. Therefore, $H$ is a subgroup of $G$.

Example. $\forall n\in\mathbb{N}\cup\{0\}$, $n\mathbb{Z}\leq(\mathbb{Z},+)$. Here, $n\mathbb{Z}=\{nx: x\in\mathbb{Z}\}$. In fact, they are all the subgroups of $(\mathbb{Z},+)$.

Definition. The cyclic subgroup of $G$ generated by $a\in G$ is $\{a^k: k\in\mathbb{Z}\}$. It is denoted by $\langle a\rangle$.

Example. Let $G$ be a group and $a\in G$. Let $C(a)=\{g\in G: ga=ag\}$. Then $C(a)\leq G$. $C(a)$ is called the centralizer of $a$ in $G$.

Example. Let $G$ be a group and let $Z(G)=\{z\in G: zx=xz\ \forall x\in G\}$. Then $Z(G)$ is a subgroup of $G$, called the center of $G$.

Let $G$ be a group and $H\leq G$. $\forall a\in G$, $a^{-1}Ha=\{a^{-1}ha: h\in H\}$ is also a subgroup of $G$.

Lemma. Suppose that $(G,\cdot)$ is a group and $H$ is a finite nonempty subset of $H$. If $H$ is closed under $\cdot$, then $H$ is a subgroup of $G$.

Proof. Let $H=\{a_1,a_2,\cdots,a_n\}$. Then $\forall a\in H$, $aH=\{aa_1,aa_2,\cdots,aa_n\}\subset H$. On the other hand, if $aa_i=aa_j$ then $a_i=a_j$. So, $|aH|=|H|$. This implies that $aH=H$ and hence
\begin{align*}
a\in aH &\Longrightarrow a=aa_i\ \mbox{for some}\ i\\
&\Longrightarrow a_i=e\in H.
\end{align*}
Since $e\in aH$, $e=aa_j$ for some $j$ $\Longrightarrow$ $a_j=a^{-1}$.

Example. The symmetric group $S_3=\{1,(123),(132),(23),(13),(12)\}$ has 5 subgroups
$\{1\}$, $\{1,(12)\}$, $\{1,(13)\}$, $\{1,(23)\}$, and $A_3=\{1,(123),(132)\}$. $A_3$ is called alternating group of degree 3. Cycles of length 2 such as (12), (13), (23) are called transpositions. Any permutation can be written as a product of either an even number of transpositions or an odd number of transpositions. If a permutation is a product of an even number of transpositions, the permutation is called an even permutation. If a permutation is a product of an odd number of transpositions, the permutation is called an odd permutation. The identity permutation 1 is an even permutation as $\begin{pmatrix} 1 & 2 & 3\\ 1 & 2 & 3 \end{pmatrix}$ can be written as (12)(21)(23)(32)(31)(13). The permutation (123) can be written as (12)(13). So, (123) is an even permutation. Similarly (132) is an even permutation as well. So it turns out that the alternating group $A_3$ is the set of all even permutations. In general, the set $A_n$ of all even permutations of $\{1,2,3,\cdots,n\}$ form a subgroup of the symmetric group $S_n$. Remember that $S_n$ has order $n!$. Since there are exactly the same number of even permutations and odd permutations, the alternating group $A_n$ of degree $n$ has order $\frac{n!}{2}$.

Example. $\mathrm{SL}(n,\mathbb{R})=\{A\in\mathrm{GL}(n,\mathbb{R}): \det A=1\}$ is a subgroup of the general linear group $\mathrm{GL}(n,\mathbb{R})$ of degree $n$. $\mathrm{SL}(n,\mathbb{R})$ is called the special linear group of degree $n$. The general linear group $\mathrm{GL}(n,\mathbb{R})$ is the group of isometries of $\mathbb{R}^n$. An isometry of $\mathbb{R}^n$ is a linear bijective map from $\mathbb{R}^n$ onto itself which preserves the inner product. What this means is that if $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is an isometry then for any two vectors $v,w\in\mathbb{R}^n$, $\langle Tv,Tw\rangle=\langle v,w\rangle$. The special linear group $\mathrm{SL}(n,\mathbb{R})$ is the group of all orientation preserving isometries of $\mathbb{R}^n$.

# Functional Analysis 4: Convergence, Cauchy Sequence, Completeness

The set $\mathbb{Q}$ of rational numbers is not complete (or not a continuum) since it has gaps or holes. For instance, $\sqrt{2}$ is not in $\mathbb{Q}$. On the other hand, the set $\mathbb{R}$ of real numbers has no gaps or holes, so it is complete (or is a continuum). Let $(x_n)$ be a sequence of real numbers. Suppose that $(x_n)$ converges to a real number $x$. Then by the triangle inequality, for any $m,n\in\mathbb{N}$, we have
$$|x_m-x_n|\leq |x_m-x|+|x-x_n|.$$
Hence, $\displaystyle\lim_{m,n\to\infty}|x_m-x_n|=0$, i.e. $(x_n)$ is a Cauchy sequence. Conversely, Georg Cantor introduced the completeness axiom that every Cauchy sequence of real numbers converges and defined a real number as the limit of a Cauchy sequence of rational numbers. For instance, consider the Cauchy sequence $(x_n)$ defined by
$$x_1=1,\ x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n},\ \forall n\geq 2.$$
If $(x_n)$ converges to a number $x$, it would satisfy $x^2=2$ i.e. $(x_n)$ converges to $\sqrt{2}$. There is another way to obtain the completeness of $\mathbb{R}$ by a Dedekind cut, though we are not going to delve into that here.

More generally, one can also consider a complete metric space and that is what we are going to study in this lecture.

Definition. A sequence $(x_n)$ is a metric space $(X,d)$ is said to converge or to be convergent to $x\in X$ if
$$\lim_{n\to\infty}d(x_n,x)=0.$$
$x$ is called the limit if $(x_n)$ and we write
$$\lim_{n\to\infty}x_n=x\ \mbox{or}\ x_n\rightarrow x.$$
If $(x_n)$ is not convergent, it is sad to be divergent. We can generalize the definiton of the convergence of a sequence we learned in calculus in terms of a metric as:

Definition. $\displaystyle\lim_{n\to\infty}d(x_n,x)=0$ if and only if given $\epsilon>0$ $\exists$ a positive integer $N$ s.t. $x_n\in B(x,\epsilon)$ $\forall n\geq N$.

A nonempty subset $M\subset X$ is said to be bounded if
$$\delta(M)=\sup_{x,y\in M}d(x,y)<\infty.$$

Lemma. Let $(X,d)$ be a metric space.

(a) A convergent sequence in $X$ is bounded and its limit is unique.

(b) If $x_n\rightarrow x$ and $y_n\rightarrow y$, then $d(x_n,y_n)\rightarrow d(x,y)$.

Proof. (a) Suppose that $x_n\rightarrow x$. Then one can find a positive integer $N$ such that $d(x_n,x)<1$ $\forall n\geq N$. Let $M=2\max\{d(x_1,x),\cdots,d(x_{N-1},x),1\}$. Then for all $m,n\in\mathbb{N}$,
\begin{align*}
d(x_m,x_n)&\leq d(x_m,x)+d(x,x_n)\ (\mbox{ (M3) triangle inequality)}\\
&\leq M.
\end{align*}
This means that $\delta((x_n))\leq M<\infty$ i.e. $(x_n)$ is bounded.

Suppose that $x_n\rightarrow x$ and $x_n\rightarrow y$. Then
\begin{align*}
0\leq d(x,y)&\leq d(x,x_n)+d(x_ny)\\
&\rightarrow 0
\end{align*}
as $n\to\infty$. So, $d(x,y)=0\Rightarrow x=y$ by (M1).

(b) By (M3),
$$d(x_n,y_n)\leq d(x_n,x)+d(x,y)+d(y,y_n)$$
and so we obtain
$$d(x_n,y_n)-d(x,y)\leq d(x_n)+d(y,y_n).$$
Similarly, we also obtain the inequality
$$d(x,y)-d(x_n,y_n)\leq d(x,x_n)+d(y_n,y).$$
Hence,
$$0\leq |d(x_n,y_n)-d(x,y)|\leq d(x_n,x)+d(y_n,y)\rightarrow 0$$
as $n\to\infty$.

Definition. A sequence $(x_n)\subset (X,d)$ is said to be Cauchy if given $\epsilon>0$ $\exists$ a positive integer $N$ such that
$$d(x_m,x_n)<\epsilon\ \forall m,n\geq N.$$
The space $X$ is said to be complete if every Cauchy sequence in $X$ converges.

Examples. The real line $\mathbb{R}$ and the complex plane $\mathbb{C}$ are complete.

Theorem. Every convergent sequence is Cauchy.

Proof. Suppose that $x_n\rightarrow x$. Then given $\epsilon>0$ $\exists$ a poksitive integer $N$ s.t. $d(x_n,x)<\frac{\epsilon}{2}$ for all $n\geq N$. Now, $\forall m,n\geq N$
$$d(x_m,x_n)\leq d(x_m,x)+d(x,x_n)<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$$
Therefore, $(x_n)$ is Cauchy.

Theorem. Let $M$ be a nonempty subset of a metric space $(X,d)$. Then

(a) $x\in\bar M\Longleftrightarrow \exists$ a seqence $(x_n)\subset M$ such that $x_n\rightarrow x$.

(b) $M$ is closed $\Longleftrightarrow$ given a sequence $(x_n)\subset M$, $x_n\rightarrow x$ implies $x\in M$.

Proof. (a) ($\Longrightarrow$) Since $x\in\bar M$, $\forall n\in\mathbb{N}$ $\exists x_n\in B\left(x,\frac{1}{n}\right)\cap M\ne\emptyset$. Let $\epsilon>0$ be given. Then by the Archimedean property, $\exists$ a positive integer $N$ s.t. $N\geq\frac{1}{\epsilon}$. Now,
$$n\geq N\Longrightarrow d(x_n,x)<\frac{1}{n}\leq\frac{1}{N}<\epsilon.$$

($\Longleftarrow$) Suppose that $\exists$ a sequence $(x_n)\subset M$ s.t. $x_n\rightarrow x$. Then given $\epsilon>0$ $\exists$ a positive integer $N$ s.t. $x_n\in B(x,\epsilon)$ $\forall n\geq N$. This means that $\forall\epsilon>0$, $B(x,\epsilon)\cap M\ne\emptyset$. So, $x\in\bar M$.

(b) ($\Longrightarrow$) Clear

($\Longleftarrow$) It suffices to show that $\bar M\subset M$. Let $x\in\bar M$. Then $\exists$ a sequence $(x_n)\subset M$ such that $x_n\rightarrow x$. By assumption, $x\in M$.
Theorem. A subspace $M$ of a complete metric space $X$ itself is complete if and only if $M$ is closed in $X$.

Proof. ($\Longrightarrow$) Let $M\subset X$ be complete. Let $(x_n)$ be a sequence in $M$ such that $x_n\rightarrow x$. Then $(x_n)$ is Cauchy. Since $M$ is complete, every Cauchy sequence must converge and hence $x\in M$. This means that $M$ is closed.

($\Longleftarrow$) Suppose that $M\subset X$ is closed. Let $(x_n)$ be a Cauchy sequence in $M\subset X$. Since $X$ is complete, $\exists x\in X$ such that $x_n\rightarrow x$. Since $M$ is closed, $x\in M$. Therefore, $M$ is complete.

Theorem. A mapping $T: X\longrightarrow Y$ is continuous at $x_0\in X$ if and only if $x_n\rightarrow x$ implies $Tx_n\rightarrow Tx_0$.

Proof. ($\Longrightarrow$) Suppose that $T$ is continuous and $x_n\rightarrow x$ in $X$. Let $\epsilon>0$ be given. Then $\exists\delta>0$ s.t. whenever $d(x,x_0)<\delta$, $d(Tx,Tx_0)<\epsilon$. Since $x_n\rightarrow x$, $\exists$ a positive integer $N$ s.t. $d(x_n,x_0)<\delta$ $\forall n\geq N$. So, $\forall n\geq N$, $d(Tx_n,Tx_0)<\epsilon$. Hence, $Tx_n\rightarrow Tx_0$.

($\Longleftarrow$) Suppose that $T$ is not continuous. Then $\exists\epsilon>0$ s.t. $\forall\delta>0$, $\exists x\ne x_0$ satisfying $d(x,x_0)<\delta$ but $d(Tx,tx_0)\geq\epsilon$. So, $\forall n=1,2,\cdots$, $\exists x_n\ne x_0$ satisfying $d(x_n,x_0)<\frac{1}{n}$ but $d(Tx_n,Tx_0)\geq\epsilon$.

# Group Theory 4: Examples of Groups

In the first lecture here, we defined a group. We have also seen an example of a group here, which is a symmetric group. In this lecture, we study some well-known examples of groups of finite and infinite orders. Recall that the order of a group os the number of elements in the group.

Examples of Groups

1. $(\mathbb{Z},+)$, $(\mathbb{Q},+)$, $(\mathbb{R},+)$, and $(\mathbb{C},+)$ are abelian groups of infinite order.
2. $(\mathbb{Q}\setminus\{0\},\cdot)$ is an abelian group of infinite order.
3. $(\mathbb{R}^+,\cdot)$ is an abelian group of infinite order. Here $\mathbb{R}^+$ denotes the set of all positive real numbers.
4. Let $E_n=\{e^{\frac{2k\pi i}{n}}: k=0,1,2,\cdots,n-1\}$. $e^{\frac{2k\pi i}{n}}$, $k=0,1,2,\cdots,n-1$ are the $n$-th roots of unity i.e. the zeros of $z^n=1$. $E_n$ forms an abelian group of order $n$. It is generated by a single element $e^{\frac{2\pi i}{n}}$. Such a group is called a cyclic group.

Definition. A group $G$ is said to be a finite group if it has a finite number of elements. The number of elements in $G$ is called the order of $G$ as mentioned previously and it is denoted by $|G|$ or $\mathrm{ord}(G)$. We use the notation $|G|$ for the order of the group $G$.

Example.Let $\mathbb{Z}_n=\{0,1,2,\cdots,n-1\}$. $\mathbb{Z}_n$ is the set of all possible remainders when an integer is divided by 2. The addition $+$ on $\mathbb{Z}_n$ is defined naturally as follows.
\begin{aligned} \mathbb{Z}_2\\ \begin{array}{|c||c|c|} \hline + & 0 & 1\\ \hline \hline 0 & 0 & 1\\ \hline 1 & 1 & 0\\ \hline \end{array} \end{aligned}
\begin{aligned} \mathbb{Z}_3\\ \begin{array}{|c||c|c|c|} \hline + & 0 & 1 & 2\\ \hline \hline 0 & 0 & 1 & 2\\ \hline 1 & 1 & 2 & 0\\ \hline 2 & 0 & 1 & 2\\ \hline \end{array} \end{aligned}
$(\mathbb{Z}_n,+)$ is an abelian group of order $n$. We will see later that the group $E_n$ and $\mathbb{Z}_n$ are indeed the same group.

Example. The set $M_{m\times n}(\mathbb{R})$ of all $m\times n$ matrices of real entries under matrix addition is an abelian group of infinite order.

Example. The set $\mathrm{GL}(n,\mathbb{R})$ of alll $n\times n$ non-singular matrices of real entries under matrix multiplication is a non-abelian group of infinite order. $\mathrm{GL}(n,\mathbb{R})$ is called the general linear group of degree $n$. It can be viewed as the set of all invertible linear transformations $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ that preserve the standard Euclidean inner product. A map $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is said to be an isometry if it is a linear isomorphism (i.e linear, one-to-one and onto, but remember from linear algebra that a linear map $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is one-to-one, it is also onto) and it preserves inner product i.e. for any vectors $v$ and $w$ in $\mathbb{R}$, $\langle Tv,Tw\rangle=\langle v,w\rangle$.

Example. [Klein four-group (Vierergruppe in German)]

Klein four-group

Let $V=\{e,a,b,c\}$ where $a$ is counterclockwise rotation of the rectangle in the picture about the $x$-axis by $180^\circ$, $b$ is counterclockwise rotation of the rectangle about the $y$-axis by $180^\circ$ and $c$ is counterclockwise rotation of the rectangle about the $z$-axis (the axis coming out of the origin toward you) by $180^\circ$. Then $a$, $b$ and $c$ satisfy the following relationship
\begin{align*}
a^2=b^2=c^2=e,\ ab&=ba=c,\ bc=cb=a,\\
ca&=ac=b.
\end{align*}
Here, $\cdot$ is the function composition i.e. successive application of rotations. By labling the vertices of the rectangle as 1, 2, 3, 4 as seen in the picture, we can define each rotation as a permutation of $\{1,2,3,4\}$.
\begin{align*}
e&=\begin{pmatrix}
1 & 2 & 3 & 4\\
1 & 2 & 3 & 4
\end{pmatrix},\ a=\begin{pmatrix}
1 & 2 & 3 & 4\\
2 & 1 & 4 &3
\end{pmatrix},\\
b&=\begin{pmatrix}
1 & 2 & 3 & 4\\
4 & 3 & 2 & 1
\end{pmatrix},\ c=\begin{pmatrix}
1 & 2 & 3 & 4\\
3 & 4 & 1 &2
\end{pmatrix}.
\end{align*}
For instance, we calculate $ab$:
\begin{align*}
ab&=\begin{pmatrix}
1 & 2 & 3 & 4\\
2 & 1 & 4 &3
\end{pmatrix}\begin{pmatrix}
1 & 2 & 3 & 4\\
4 & 3 & 2 & 1
\end{pmatrix}\\
&=\begin{pmatrix}
1 & 2 & 3 & 4\\
2 & 1 & 4 &3
\end{pmatrix}\begin{pmatrix}
4 & 3 & 2 & 1\\
3 & 4 & 1 & 2
\end{pmatrix}\\
&=\begin{pmatrix}
1 & 2 & 3 & 4\\
3 & 4 & 1 &2
\end{pmatrix}\\
&=c.
\end{align*}

Example. [The $n$-th dihedral group $D_n$, $n\geq 3$]

Consider an equilateral triangle shown in the following picture.

Dihedral group D3

$\rho$ is counterclockwise rotation about the axis coming out of the origin by $\frac{360^\circ}{3}=120^\circ$ and $\mu_i$, $i=1,2,3$ is counterclockwise rotation about the axis of rotation through each vertex $i$ by $180^\circ$. As permutations of the vertices $\{1,2,3\}$, $\rho$ and $\mu_i$ , $i=1,2,3$ are given by
\begin{align*}
\rho&=\begin{pmatrix}
1 & 2 & 3\\
2 & 3 & 1
\end{pmatrix},\ \mu_1=\begin{pmatrix}
1 & 2 & 3\\
1 & 3 & 2
\end{pmatrix}\\
\mu_2&=\begin{pmatrix}
1 & 2 & 3\\
3 & 2 & 1
\end{pmatrix},\ \mu_3=\begin{pmatrix}
1 & 2 & 3\\
2 & 1 &3
\end{pmatrix}.
\end{align*}
$D_3=\{e,\rho,\rho^2,\mu_1,\mu_2,\mu_3\}$ is a group called the 3rd dihedral group. $$\mu_1\mu_2=\rho^2\ne\rho=\mu_2\mu_1.$$
So, we see that $D_3$ is not an abelian group. $D_3$ is the group of symmetries of an equilateral triangle.

Now this time let us consider a square as shown in the following picture.

Dihedral group D4

$\rho$ is counterclockwise rotation about the $z$-axis coming out of the origin by $\frac{360^\circ}{4}=90^\circ$. $\mu_i$, $i=1,2$ are counterclockwise rotations about $y$-axis and $x$-axis, respectively by $180^\circ$. $\delta_i$, $i=1,2$ are counterclockwise rotations about the axis through the vertices 2 and 4 and the vertices through 1 and 3, respectively by $180^\circ$. As permutations of the vertices $\{1,2,3,4\}$, $\rho$, $\mu_i$ and $\delta_i$, $i=1,2$ are given by
\begin{align*}
\rho&=\begin{pmatrix}
1 & 2 & 3 & 4\\
2 & 3 & 4 & 1
\end{pmatrix},\\
\mu_1&=\begin{pmatrix}
1 & 2 & 3 & 4\\
2 & 1 & 4 & 3
\end{pmatrix},\ \mu_2=\begin{pmatrix}
1 & 2 & 3 & 4\\
4 & 3 & 2 & 1
\end{pmatrix},\\
\delta_1&=\begin{pmatrix}
1 & 2 & 3 & 4\\
3 & 2 & 1 & 4
\end{pmatrix},\ \delta_2=\begin{pmatrix}
1 & 2 & 3 & 4\\
1 & 4 & 3 & 2
\end{pmatrix}.
\end{align*}
$D_4=\{e,\rho,\rho^2,\rho^3,\mu_1,\mu_2,\delta_1,\delta_2\}$ is the group of symmetries of a square, called the 4th dihedral group. It is also called the octic group. Note that $|D_n|=2n$, $n\geq 3$.

# Functional Analysis 3: Basic (Metric) Topology

Let $(X,d)$ be a metric space.

Definition. A subset $U\subset X$ is said to be open if $\forall x\in U$ $\exists\epsilon>0$ s.t. $B(x,\epsilon)\subset U$.

If $U\subset X$ is open then $U$ can be expressed as union of open balls $B(x,\epsilon)$. Hence, the set of all open balls in $X$, $\mathcal{B}=\{B(x,\epsilon): x\in X,\ \epsilon>0\}$ form a basis for a topology (a metric topology, the topology induced by the metric $d$) on $X$. Those who have not studied topology before may simply understand it as the set of all open sets in $X$.

Definition. A subset $F\subset X$ is said to be closed if its complement, $F^c=X\setminus F$ is open in $X$.

The following is the definition of a continuous function that you are familiar with from calculus. The definition is written in terms of metrics.

Definition. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $T:X\longrightarrow Y$ is said to be continuous at $x_0\in X$ if $\forall\epsilon>0$ $\exists\delta>0$ s.t $d_Y(Tx,Tx_0)<\epsilon$ whenever $d_X(x,x_0)<\delta$.

$T$ is said to be continuous if it is continuous at every point of $X$.

The above definition can be generalized in terms of open sets as follows.

Theorem. A mapping $T: (X,d_X)\longrightarrow(Y,d_Y)$ is continuous if and only if $\forall$ open set $U$ in $Y$, $T^{-1}U$ is open in $X$.

Proof. (Only if, $\Rightarrow$) Suppose that $T:X\longrightarrow Y$ is continuous. Let $U$ be open in $Y$. Then we show that $T^{-1}U$ is open in $X$. Let $x_0\in T^{-1}U$. Then $Tx_0\in U$. Since $U$ is open in $Y$, $\exists\epsilon>0$ s.t. $B(Tx_0,\epsilon)\subset U$. By the continuity of $T$, for this $\epsilon>0$ $\exists\delta>0$ s.t. whenever $d(x,x_0)<\delta$, $d(Tx,Tx_0)<\epsilon$. This means that
$$TB(x_0,\delta)\subset B(Tx_0,\epsilon)\subset U\Longrightarrow B(x_0,\delta)\subset T^{-1}(TB(x_0,\delta))\subset T^{-1}U.$$ Hence, $T^{-1}U$ is open in $X$.

(If, $\Leftarrow$) Suppose that $\forall$ open set $U$ in $Y$, $T^{-1}U$ is open in $X$. We show that $T$ is continuous. Let $x_0\in X$ and let $\epsilon>0$ be given. Then $B(Tx_0,\epsilon)$ is open in $Y$. So by the assumption, $x_0\in T^{-1}B(Tx_0,\epsilon)$ is open in $X$. This means that $\exists\delta>0$ s.t.
$$B(x_0,\delta)\subset T^{-1}B(Tx_0,\epsilon)\Longrightarrow TB(x_0,\delta)\subset T(T^{-1}B(Tx_0,\epsilon))\subset B(Tx_0,\epsilon).$$ This is equivalent to saying that $\exists\delta>0$ s.t. whenever $d(x,x_0)<\delta$, $d(Tx,Tx_0)<\epsilon$. That is, $T$ is continuous at $x_0$. Since the choice $x_0\in X$ was arbitrary, the proof is complete.

Let $A\subset X$. $x\in X$ is called an accumulation point or a limit point of $A$ if $\forall$ open set $U(x)$ in $X$, $(U(x)-\{x\})\cap A\ne\emptyset$. Here the notation $U(x)$ means that it contains $x$. The set of all accumulation points of $A$ is denoted by $A’$ and is called the derived set of $A$. $\bar A:=A\cup A’$ is called the closure of $A$. $\bar A$ is the smallest closed set containing $A$.

Theorem. Let $A\subset X$. Then $x\in\bar A$ if and only if $\forall$ open set $U(x)$, $U(x)\cap A\ne\emptyset$.

Definition. $D\subset X$ is said to be dense if $\bar D=X$. This means that $\forall$ open set $U$ in $X$, $U\cap D\ne\emptyset$.

Definition. $X$ is said to be separable if it has a countable dense subset.

Examples. The real line $\mathbb{R}$ is separable. The complex plane $\mathbb{C}$ is also separable.

Theorem. The space $\ell^\infty$ is not separable.

Proof. Let $y=(\eta_1,\eta_2,\eta_3,\cdots)$ be a sequence of zeros and ones. Then $y\in\ell^\infty$. We can then associate $y$ with the binary representation
$$\hat y=\frac{\eta_1}{2}+\frac{\eta_2}{2^2}+\frac{\eta_3}{2^3}+\cdots\in [0,1].$$ Each $\hat y\in [0,1]$ has a binary representation and different $\hat y$’s have different binary representations. So, there are uncountably many sequences of zeros and ones. If $y$ and $z$ are sequences of zeros and ones and $y\ne z$, then $d(y,z)=1$. This means that for any two distinct sequences $y$ and $z$ of zeros and ones, $B\left(y,\frac{1}{3}\right)\cap B\left(z,\frac{1}{3}\right)=\emptyset$. Let $A$ be a dense subset of $\ell^\infty$. Then for each sequence $y$ of zeros and ones, $B\left(y,\frac{1}{3}\right)$ has at least one element of $A$. This means that $A$ cannot be countable.

Theorem. The space $\ell^p$ with $1\leq p<\infty$ is separable.

Proof. Let $A$ be the set of all sequences $y$ of the form
$$y=(\eta_1,\eta_2,\cdots,\eta_n,0,0,\cdots,0),$$ where $n$ is a positive integer and the $\eta_j$’s are rational. For each $n=1,2,\cdots$, the number of sequences of the form $y=(\eta_1,\eta_2,\cdots,\eta_n,0,0,\cdots,0)$ is the same as the number of functions from $\{1,2,3,\cdots,n\}$ to $\mathbb{Q}$, the set of all rational numbers. $\mathbb{Q}$ has the cardinality $\aleph_0$ and so the number is $\aleph_0^n=\aleph_0$. The cardinality of $A$ is then $\aleph_0\cdot\aleph_0=\aleph_0$ i.e. $A$ is countable. Now we show that $A$ is dense in $\ell^p$. Let $x=(\xi_j)\in\ell^p$. Let $\epsilon>0$ be given. Since $\displaystyle\sum_{j=1}^\infty|\xi_j|^p<\infty$, $\exists$ a positive integer $N$ s.t. $\displaystyle\sum_{j=N+1}^\infty|\xi_j|^p<\frac{\epsilon^p}{2}$. Since rationals are dense in $\mathbb{R}$, one can find $y=(\eta_1,\eta_2,\cdots,\eta_N,0,0,\cdots)\in A$ s.t. $\displaystyle\sum_{j=1}^N|\xi_j-\eta_j|^p<\frac{\epsilon^p}{2}$. Hence,
$$[d(x,y)]^p=\sum_{j=1}^N|\xi_j-\eta_j|^p+\sum_{j=N+1}^\infty|\xi_j|^p<\epsilon^p,$$
i.e. $d(x,y)<\epsilon$. This means that $y\in B(x,\epsilon)\cap A\ne\emptyset$. This completes the proof.

# Group Theory 3: Preliminaries (Basic Number Theory)

In this lecture, we study some basic number theory as it is needed to study group theory.

Let $\mathbb{Z}$ denote the set of integers. $\mathbb{Z}$ satisfies well-ordering principle, namely any non-empty set of nonnegative integers has a smallest member.

One of the most fundamental theorems regarding numbers is Euclid’s Algorithm. Although we will not discuss its proof, it can be proved using well-ordering principle.

Theorem. [Euclid's Algorithm] If $m$ and $n$ are integers with $n>0$, then $\exists$ integers $q$ and $r$ with $0\leq r<n$ such that $m=qn+r$.

Euclid’s algorithm hints us how we can define the notion that one integer divides another.

Definition. Given $m\ne 0, n\in\mathbb{Z}$, we say $m$ divides $n$ and write $m|n$ if $n=cm$ for some $c\in\mathbb{Z}$.

Examples. $2|14$, $(-7)|14$, $4|(-16)$.

If $m|n$, we call $m$ a divisor or a factor of $n$, and $n$ a multiple of $m$. To indicate $m$ is not a divisor of $n$, we write $m\not|n$. For example, $3\not|5$.

Lemma. The following properties hold.

(a) $1|n$ $\forall n$.

(b) If $m\ne 0$ then $m|0$.

(c) If $m|n$ and $n|q$, then $m|q$.

(d) If $m|n$ and $m|q$ then $m|(\mu n+\nu q)$ $\forall \mu,\nu$.

(e) If $m|1$ then $m=\pm 1$.

(f) If $m|n$ and $n|m$ then $m=\pm n$.

Definition. Given $a,b$ (not both 0), their greatest common divisor (in short gcd) $c$ is defined by the following properties:

(a) $c>0$

(b) $c|a$ and $c|b$

(c) If $d|a$ and $d|b$ then $d|c$.

If $c$ is the gcd of $a$ and $b$, we write $c=(a,b)$.

$(24,9)=3$. Note that the gcd 3 can be written in terms of 24 and 9 as $3\cdot 9+1\cdot (-24)$ or $(-5)9+2\cdot 24$. In general, we have the following theorem holds.

Theorem. If $a,b$ are not both 0, their gcd exists uniquely. Moreover, $\exists m,n\in\mathbb{Z}$ s.t. $c=ma+nb$.

Now let us talk about how to find the gcd of two positive numbers $a$ and $b$. W.L.O.G. (Without Loss Of Generality), we may assume that $b<a$. Then by Euclid’s algorithm we have
$$a=bq+r,\ \mbox{where}\ 0\leq r<b.$$
Let $c=(a,b)$. Then $c|r$, so $c$ is a common divisor of $b$ and $r$. If $d$ is a common divisor of $b$ and $r$, it is also a common divisor of $a$ and $b$. This implies that $d\leq c$ and so $c=(b,r)$. Finding $(b,r)$ is of course easier because one of the numbers is smaller than before.

Example. [Finding GCD]
\begin{aligned} (100,28)&=(28,16)\ &(100&=28\cdot 3+16)\\ &=(16,12)\ &(28&=16\cdot 1+12)\\ &=(12,4)\ &(16&=12\cdot 1+14)\\ &=4. \end{aligned}
By working backward, we can also find integers $m$ and $n$ such that
$$4=m\cdot 100+n\cdot 28.$$
\begin{align*}
4&=16+12(-1)\\
&=16+(-1)[28+(-1)16]\\
&=(-1)28+2\cdot 16\\
&=(-1)28+2[100+(-3)28]\\
&=2\cdot 100+(-7)28.
\end{align*}
Therefore, $m=2$ and $n=-7$.

Definition. We say that $a$ and $b$ are relatively prime if $(a,b)=1$.

Theorem. The integers $a$ and $b$ are relatively prime if and only if $1=ma+nb$ for some $m$ and $n$.

Theorem. If $(a,b)=1$ and $a|bc$ then $a|c$.

Theorem. If $b$ and $c$ are both relatively prime to $a$, then $bc$ is also relatively prime to $a$.

Definition. A prime number, or shortly prime, is an integer $p>1$ such that $\forall a\in\mathbb{Z}$, either $p|a$ or $(p,a)=1$.

Suppose that $p$ is a prime as defined above and $p=ab$, where $1\leq a<p$. Then $p\not|a$ since $a<p$, so $(p,a)=1$. This implies that $p|b$. On the other hand, $b|p(=ab)$ and hence $p=b$ and $a=1$. So, the above definition coincides with the definition of a prime we are familiar with.

Theorem. If $p$ is a prime and $p|a_1a_2\cdots a_n$, then $p|a_i$ for some $i$ with $1\leq i\leq n$.

Proof. If $p|a_1$, we are done. If not, $(p,a_1)=1$ and so $p|a_2a_3\cdots a_n$. Continuing this, we see that $p|a_i$ for some $i$.

Regarding primes, we have the following theorems.

Theorem. If $n>1$, then either $n$ is a prime or the product of primes.

Theorem. [Unique Factorization Theorem] Given $n>1$, there is a unique way to write $n$ in the form $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_1<p_2<\cdots<p_k$ are primes and the exponents $a_1,\cdots,a_k$ are all positive.

Theorem. [Euclid] There is an infinite number of primes.

# Functional Analysis 2: $\ell^p$ and $L^p$ as Metric Spaces

Let $p\geq 1$ be a fixed number and let
$$\ell^p=\left\{x=(\xi_j): \sum_{j=1}^\infty|\xi_j|^p<\infty\right\}.$$
Define $d:\ell^p\times\ell^p\longrightarrow\mathbb{R}^+\cup\{0\}$ by
$$d(x,y)=\left(\sum_{j=1}^\infty|\xi_j-\eta_j|^p\right)^{\frac{1}{p}}.$$
Then $(\ell^p,d)$ is a metric space. The properties (M1) and (M2) are clearly satisfied. We prove the remaining property (M3) the triangle inequality. $p=1$ case can be easily shown by the triangle inequality of numbers. We need a few steps to do this. First we prove the following inequality: $\forall\alpha>0,\beta>0$,
$$\alpha\beta\leq\frac{\alpha^p}{p}+\frac{\beta^q}{q},$$
where $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$. The numbers $p$ and $q$ are called conjugate exponenets. It follows from $\frac{1}{p}+\frac{1}{q}=1$ that $(p-1)(q-1)=1$ i.e. $\frac{1}{p-1}=q-1$. If we let $u=t^{p-1}$ then $t=u^{\frac{1}{p-1}}=u^{q-1}$. By comparing areas, we obtain
$$\alpha\beta\leq\int_0^{\alpha}t^{p-1}dt+\int_0^{\beta}u^{q-1}du=\frac{\alpha^p}{p}+\frac{\beta^q}{q}.$$
Next, using this inequality we prove the Hölder inequality
$$\sum_{j=1}^\infty|\xi_j\eta_j|\leq\left(\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}\left(\sum_{m=1}^\infty|\eta_m|^q\right)^{\frac{1}{q}}$$
where $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$. When $p=2$ and $q=2$, we obtain the well-known Cauchy-Schwarz inequality.

Proof. Let $(\tilde\xi_j)$ and $(\tilde\eta_j)$ be two sequences such that
$$\sum_{j=1}^\infty|\tilde\xi_j|^p=1,\ \sum_{j=1}^\infty|\tilde\eta_j|^q=1.$$
Let $\alpha=|\tilde\xi_j|$ and $\beta=|\tilde\eta_j|$. Then by the inequality we proved previously,
$$|\tilde\xi_j\tilde\eta_j|\leq\frac{|\tilde\xi_j|^p}{p}+\frac{|\tilde\eta_j|^q}{q}$$
and so we obtain
$$\sum_{j=1}^\infty|\tilde\xi_j\tilde\eta_j|\leq\sum_{j=1}^\infty\frac{|\tilde\xi_j|^p}{p}+\sum_{j=1}^\infty\frac{|\tilde\eta_j|^q}{q}=1.$$
Now take any nonzero $x=(\xi_j)\in\ell^p$, $y=(\eta_j)\in\ell^q$. Setting
$$\tilde\xi_j=\frac{\xi_j}{\left(\displaystyle\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}},\ \tilde\eta_j=\frac{\eta_j}{\left(\displaystyle\sum_{m=1}^\infty|\eta_m|^q\right)^{\frac{1}{q}}}.$$
results the Hölder inequality.

Next, we prove the Minkowski inequality
$$\left(\sum_{j=1}^\infty|\xi_j+\eta_j|^p\right)^{\frac{1}{p}}\leq\left(\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}+\left(\sum_{m=1}^\infty|\eta_m|^p\right)^{\frac{1}{p}}$$
where $x=(\xi_j)\,y=(\eta_j)\in\ell^p$ and $p\geq 1$. $p=1$ case comes from the triangle inequality for numbers. Let $p>1$. Then
\begin{align*}
|\xi_j+\eta_j|^p&=|\xi_j+\eta_j||\xi_j|\eta_j|^{p-1}\\
&=(|\xi_j|+|\eta_j|)|\xi_j+\eta_j|^{p-1}\ (\mbox{triangle inequality for numbers}).
\end{align*}
For a fixed $n$, we have
$$\sum_{j=1}^n|\xi_j+\eta_j|^p\leq\sum_{j=1}^n|\xi_j||\xi_j+\eta_j|^{p-1}+\sum_{j=1}^n|\eta_j||\xi_j+\eta_j|^{p-1}.$$
Using the Hölder inequality, we get the following inequality
\begin{align*}
\sum_{j=1}^n|\xi_j||\xi_j+\eta_j|^{p-1}&\leq \sum_{j=1}^\infty |\xi_j||\xi_j+\eta_j|^{p-1}\\
&\leq\left(\sum_{k=1}^\infty |\xi_k|^p\right)^{\frac{1}{p}}\left(\sum_{m=1}^\infty(|\xi_m+\eta_m|^{p-1})^q\right)^{\frac{1}{q}}\ (\mbox{Hölder})\\
&=\left(\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}\left(\sum_{m=1}^\infty|\xi_m+\eta_m|^p\right)^{\frac{1}{q}}.
\end{align*}
Similarly we also get the inequality
$$\sum_{j=1}^n|\eta_j||\xi_j+\eta_j|^{p-1}\leq \left(\sum_{k=1}^\infty|\eta_k|^p\right)^{\frac{1}{p}}\left(\sum_{m=1}^\infty|\xi_m+\eta_m|^p\right)^{\frac{1}{q}}.$$
Combining these two inequalities, we get
$$\sum_{j=1}^n|\xi_j+\eta_j|^p\leq\left\{\left(\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}+\left(\sum_{k=1}^\infty|\eta_k|^p\right)^{\frac{1}{p}}\right\}\left(\sum_{m=1}^\infty|\xi_m+\eta_m|^p\right)^{\frac{1}{q}}$$
and by taking the limit $n\to \infty$ on the left hand side, we get
$$\sum_{j=1}^\infty|\xi_j+\eta_j|^p\leq\left\{\left(\sum_{k=1}^\infty|\xi_k|^p\right)^{\frac{1}{p}}+\left(\sum_{k=1}^\infty|\eta_k|^p\right)^{\frac{1}{p}}\right\}\left(\sum_{m=1}^\infty|\xi_m+\eta_m|^p\right)^{\frac{1}{q}}.$$
Finally dividing this inequality by $\displaystyle\left(\sum_{m=1}^\infty|\xi_m+\eta_m|^p\right)^{\frac{1}{q}}$ results the Minkowski inequality. The Minkowski inequality tells that
$$d(x,y)=\left(\sum_{j=1}^\infty|\xi_j-\eta_j|^p\right)^{\frac{1}{p}}<\infty$$
for $x,y\in\ell^p$. Let $x=(\xi_j), y=(\eta_j),\ z=(\zeta_j)\in\ell^p$. Then
\begin{align*}
d(x,y)&=\left(\sum_{j=1}^\infty|\xi_j-\eta_j|^p\right)^{\frac{1}{p}}\\
&\leq\left(\sum_{j=1}^\infty[|\xi_j-\zeta_j|+|\zeta_j-\eta_j|]^p\right)^{\frac{1}{p}}\\
&\leq\left(\sum_{j=1}^\infty|\xi_j-\zeta_j|^p\right)^{\frac{1}{p}}+\left(\sum_{j=1}^\infty|\zeta_j-\eta_j|^p\right)^{\frac{1}{p}}\\
&=d(x,z)+d(z,y).
\end{align*}

A measurable function $f$ on a closed interval $[a,b]$ is said to belong to $L^p$ if $\int_a^b|f(t)|^p dt<\infty$. $L^p$ is a vector space. For functions $f,g\in L^p$, we define
$$d(f,g)=\left\{\int_a^b|f(t)-g(t)|^pdt\right\}^{\frac{1}{p}}.$$
Then clearly (M2) symmetry is satisfied and one can also prove that (M3) triangle inequality holds. However, (M1) is not satisfied since what we have is that if $d(f,g)=0$ then $f=g$ a.e. (almost everywhere) i.e. the set $\{t\in[a,b]: f(t)\ne g(t)\}$ has measure $0$. It turns out that $=$ a.e. is an equivalence relation on $L^p$, so by considering $f\in L^p$ as its equivalence class $[f]$, $d$ can be defined as a metric on $L^p$ (actually the quotient space of $L^p$). Later, we will be particularly interested in the case when $p=2$ in which case $L^p$ as well as $\ell^p$ become Hilbert spaces. Those of you who want to know details about $L^p$ space are referred to

Real Analysis, H. L. Royden, 3rd Edition. Macmillan Publishing Company, 1988

# Group Theory 2: Preliminaries (Functions)

In my previous notes here, I mentioned some about logical symbols. The logical symbols I will use often are $\forall$ which means “for all”, “for any”, “for each”, or “for every” depending on the context, $\exists$ which means “there exists”, and $\ni$ which means “such that” (don’t be confused with $\in$ which means “be an element of”). We also use s.t. for “such that.” There are also $\Longrightarrow$ which means “implies” and $\Longleftrightarrow$ which means “if and only if.” I guess these pretty much cover what we use most of time.

Now lets review about functions in a more formal way. Let $X$ and $Y$ be two non-empty sets. The the Cartesian product $X\times Y$ of $X$ and $Y$ is defined as the set
$$X\times Y=\{(x,y): x\in X,\ y\in Y\}.$$
A subset $f$ of the Cartesian product $X\times Y$ (we write $f\subset X\times Y$) is called a graph from $X$ to $Y$. A graph $f\subset X\times Y$ is called a function from $X$ to $Y$ (we write $f: X\longrightarrow Y$) if whenever $(x,y_1),(x,y_2)\in f$, $y_1=y_2$. If $f: X\longrightarrow Y$ and $(x,y)\in f$, we also write $y=f(x)$. A function $f: X\longrightarrow Y$ is said to be one-to-one or injective if whenever $(x_1,y),(x_2,y)\in f$, $x_1=x_2$. This is equivalent to saying $f(x_1)=f(x_2)$ implies $x_1=x_2$. A function $f: X\longrightarrow Y$ is said to be onto or surjective if $\forall y\in Y$ $\exists x\in X$ s.t. $(x,y)\in f$. A function $f: X\longrightarrow Y$ is said to be one-to-one and onto (or bijective) if it is both one-to-one and onto (or both injective and surjective).

Let $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ be two functions. Then the composition or the composite function $g\circ f: X\longrightarrow Z$ is defined by $g\circ f(x)=g(f(x))$ $\forall x\in X$. The function composition $\circ$ may be considered as an operation and it is associative.

Lemma. If $h: X\longleftrightarrow Y$, $g:Y\longleftrightarrow Z$ and $f:Z\longleftrightarrow W$, then $f\circ(g\circ h)=(f\circ g)\circ h$.

Note that $\circ$ is not commutative i.e. it is not necessarily true that $f\circ g=g\circ f$ even when both $f\circ g$ and $g\circ f$ are defined.

The following lemmas will be useful when we study group theory later.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are one-to-one, so is $g\circ f: X\longrightarrow Z$.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are onto, so is $g\circ f: X\longrightarrow Z$.

As an immediate consequence of combining these two lemmas, we obtain

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are bijective, so is $g\circ f: X\longrightarrow Z$.

If $f\subset X\times Y$, then the inverse graph $f^{-1}\subset Y\times X$ is defined by
$$f^{-1}=\{(y,x)\in Y\times X: (x,y)\in f\}.$$
If $f: X\longrightarrow Y$ is one-to-one and onto (bijective) then its inverse graph $f^{-1}$ is a function $f^{-1}: Y\longrightarrow X$. The inverse $f^{-1}$ is also one-to-one and onto.

Lemma. If $f: X\longrightarrow Y$ is a bijection, then $f\circ f^{-1}=\imath_Y$ and $f^{-1}\circ f=\imath_X$, where $\imath_X$ and $\imath_Y$ are the identity mappings of $X$ and $Y$, respectively.

Let $A(X)$ be the set of all one-to-one functions of $X$ onto $X$ itself. Then $(A(X),\circ)$ is a group. If $X$ is a finite set of $n$-elements (we may conveniently say $X=\{1,2,\cdots,n\})$, then $(A(X),\circ)$ is a finite group of order $n!$, called the symmetric group of degree $n$. The symmetric group of degree $n$ is denoted by $S_n$ and the elements of $S_n$ are called permutations.

# Analytic Continuation

The function $f(z)=\displaystyle\frac{1}{1+z}$ has an isolated singularity at $z=-1$. It has the Maclaurin series representation

$$f(z)=\sum_{n=0}^\infty(-1)^nz^n$$
for $|z|<1$. The power series $f_1(z)=\displaystyle\sum_{n=0}^\infty(-1)^nz^n$ converges only on the open unit disk $D_1:\ |z|<1$. For instance, the series diverges at $z=\frac{3}{2}i$ i.e. $f_1\left(\frac{3}{2}i\right)$ is not defined. The first 25 partial sums of the series $f_1\left(\frac{3}{2}i\right)$ are listed below and they do not appear to be approaching somewhere.

S[1] = 1.
S[2] = 1. – 1.500000000 I
S[3] = -1.250000000 – 1.500000000 I
S[4] = -1.250000000 + 1.875000000 I
S[5] = 3.812500000 + 1.875000000 I
S[6] = 3.812500000 – 5.718750000 I
S[7] = -7.578125000 – 5.718750000 I
S[8] = -7.578125000 + 11.36718750 I
S[9] = 18.05078125 + 11.36718750 I
S[10] = 18.05078125 – 27.07617188 I
S[11] = -39.61425781 – 27.07617188 I
S[12] = -39.61425781 + 59.42138672 I
S[13] = 90.13208008 + 59.42138672 I
S[14] = 90.13208008 – 135.1981201 I
S[15] = -201.7971802 – 135.1981201 I
S[16] = -201.7971802 + 302.6957703 I
S[17] = 455.0436554 + 302.6957703 I
S[18] = 455.0436554 – 682.5654831 I
S[19] = -1022.848225 – 682.5654831 I
S[20] = -1022.848225 + 1534.272337 I
S[21] = 2302.408505 + 1534.272337 I
S[22] = 2302.408505 – 3453.612758 I
S[23] = -5179.419137 – 3453.612758 I
S[24] = -5179.419137 + 7769.128706 I
S[25] = 11654.69306 + 7769.128706 I

Also shown below are the graphics of partial sums of the series $f_1\left(\frac{3}{2}i\right)$.

The first 10 partial sums

The first 20 partial sums

The first 30 partial sums

Let us expand $f(z)=\displaystyle\frac{1}{1+z}$ at $z=i$. Then we obtain
\begin{align*}
f(z)&=\frac{1}{1+z}\\
&=\frac{1}{1+i}\cdot\frac{1}{1+\frac{z-i}{1+i}}\\
&=\sum_{n=0}^\infty (-1)^n\frac{(z-i)^n}{(1+i)^{n+1}}
\end{align*}
for $|z-i|<\sqrt{2}$. Let $f_2(z)=\displaystyle\sum_{n=0}^\infty (-1)^n\frac{(z-i)^n}{(1+i)^{n+1}}$. This series converges only on the open disk $D_2:\ |z-i|<\sqrt{2}$, in particular at $z=\frac{3}{2}i$ and $f_2\left(\frac{3}{2}i\right)=f\left(\frac{3}{2}i\right)=\frac{4}{13}-\frac{6}{13}i$. The first 25 partial sums of the series $f_2\left(\frac{3}{2}i\right)$ are listed below and it appears that they are approaching a number. In fact, they are approaching the complex number $f\left(\frac{3}{2}i\right)=\frac{4}{13}-\frac{6}{13}i$.

S[1] = 0.5000000000 – 0.5000000000 I
S[2] = 0.2500000000 – 0.5000000000 I
S[3] = 0.3125000000 – 0.4375000000 I
S[4] = 0.3125000000 – 0.4687500000 I
S[5] = 0.3046875000 – 0.4609375000 I
S[6] = 0.3085937500 – 0.4609375000 I
S[7] = 0.3076171875 – 0.4619140625 I
S[8] = 0.3076171875 – 0.4614257812 I
S[9] = 0.3077392578 – 0.4615478516 I
S[10] = 0.3076782227 – 0.4615478516 I
S[11] = 0.3076934814 – 0.4615325928 I
S[12] = 0.3076934814 – 0.4615402222 I
S[13] = 0.3076915741 – 0.4615383148 I
S[14] = 0.3076925278 – 0.4615383148 I
S[15] = 0.3076922894 – 0.4615385532 I
S[16] = 0.3076922894 – 0.4615384340 I
S[17] = 0.3076923192 – 0.4615384638 I
S[18] = 0.3076923043 – 0.4615384638 I
S[19] = 0.3076923080 – 0.4615384601 I
S[20] = 0.3076923080 – 0.4615384620 I
S[21] = 0.3076923075 – 0.4615384615 I
S[22] = 0.3076923077 – 0.4615384615 I
S[23] = 0.3076923077 – 0.4615384616 I
S[24] = 0.3076923077 – 0.4615384615 I
S[25] = 0.3076923077 – 0.4615384615 I

The following graphics shows that the real parts of the partial sums of the series $f_2\left(\frac{3}{2}i\right)$ are approaching $\frac{3}{14}$ (blue line).

The real parts of the first 25 partial sums

The next graphics shows that the imaginary parts of the partial sums of the series $f_2\left(\frac{3}{2}i\right)$  are approaching $-\frac{6}{13}$ (blue line).

The imaginary parts of the first 25 partial sums

Also shown below is the graphics of the first 25 partial sums of the series $f_2\left(\frac{3}{2}i\right)$. They are approaching the complex number $f\left(\frac{3}{2}i\right)=\frac{4}{13}-\frac{6}{13}i$ (the intersection of horizontal and vertical blue lines).

The first 25 partial sums

Note that $f_1(z)=f_2(z)$ on $D_1\cap D_2$. Define $F(z)$ as

$$F(z)=\left\{\begin{array}{ccc} f_1(z) & \mbox{if} & z\in D_1,\\ f_2(z) & \mbox{if} & z\in D_2. \end{array}\right.$$

Analytic continuation

Then $F(z)$ is analytic in $D_1\cup D_2$. The function $F(z)$ is called the analytic continuation into $D_1\cup D_2$ of either $f_1$ or $f_2$, and $f_1$ and $f_2$ are called elements of $F$. The function $f_1(z)$ can be continued analytically to the punctured plane $\mathbb{C}\setminus\{-1\}$ and the function $f(z)=\frac{1}{1+z}$ is indeed the analytic continuation into $\mathbb{C}\setminus\{-1\}$ of $f_1$. In general, whenever analytic continuation exists it is unique.

# Functional Analysis 1: Metric Spaces

This is the first of series of lecture notes I intend to write for a graduate Functional Analysis course I am teaching in the Fall.

What is functional analysis? Functional analysis is an abstract branch of mathematics, especially of analysis, concerned with the study of vector spaces of functions. These vector spaces of functions arise naturally when we study linear differential equations as solutions of a linear differential equation form a vector space. Functional analytic methods and results are important in various fields of mathematics (for example, differential geometry, ergodic theory, integral geometry, noncommutative geometry, partial differential equations, probability, representation theory etc.) and its applications, in particular, in economics, finance, quantum mechanics, quantum field theory, and statistical physics. Topics in this introductory functional analysis course include metric spaces, Banach spaces, Hilbert spaces, bounded linear operators, the spectral theorem, and unbounded linear operators.

While functional analysis is a branch of analysis, due to its nature linear algebra is heavily used. So, it would be a good idea to brush up on linear algebra among other things you need to study functional analysis.

In functional analysis, we study analysis on an abstract space $X$ rather than the familiar $\mathbb{R}$ or $\mathbb{C}$. In order to consider fundamental notions in analysis such as limits and convergence, we need to have distance defined on $X$ so that we can speak of nearness or closeness. A distance on $X$ can be defined as a function, called a distance function or a metric, $d: X\times X\longrightarrow\mathbb{R}^+\cup\{0\}$ satisfying the following properties:

(M1) $d(x,y)=0$ if and only if $x=y$.

(M2) $d(x,y)=d(y,x)$ (Symmetry)

(M3) $d(x,y)\leq d(x,z)+d(z,y)$ (Triangle Inequality)

Here $\mathbb{R}^+$ denotes the set of all positive real numbers. You can easily see how mathematicians came up with this definition of a metric. (M1)-(M3) are the properties that the familiar distance on $\mathbb{R}$, $d(x,y)=|x-y|$ satisfies. The space $X$ with a metric $d$ is called a metric space and we usually write it as $(X,d)$.

Example. Let $x=(\xi_1,\cdots,\xi_n), y=(\eta_1,\cdots,\eta_n)\in\mathbb{R}^n$. Define
$$d(x,y)=\sqrt{(\xi_1-\eta_1)^2+\cdots+(\xi_n-\eta_n)^2}.$$
Then $d$ is a metric on $\mathbb{R}^n$ called the Euclidean metric.

This time, let $x=(\xi_1,\cdots,\xi_n), y=(\eta_1,\cdots,\eta_n)\in\mathbb{C}^n$ and define
$$d(x,y)=\sqrt{|\xi_1-\eta_1|^2+\cdots+|\xi_n-\eta_n|^2}.$$
Then $d$ is a metric on $\mathbb{C}^n$ called the Hermitian metric. Here $|\xi_i-\eta_i|^2=(\xi_i-\eta_i)\overline{(\xi_i-\eta_i)}$.

Of course these are pretty familiar examples. If there can be only these familiar examples, there would be no point of considering abstract space. In fact, the abstraction allows to discover other examples of metrics that are not so intuitive.

Example. Let $X$ be the set of all bounded sequences of complex numbers
$$X=\{(\xi_j): \xi_j\in\mathbb{C},\ j=1,\cdots\}.$$
For $x=(\xi_j), y=(\eta_j)\in X$, define
$$d(x,y)=\sup_{j\in\mathbb{N}}|\xi_j-\eta_j|.$$
Then $d$ is a metric on $X$. The metric space $(X,d)$ is denoted by $\ell^\infty$.

Example. Let $X$ be the set of continuous real-valued functions define on the closed interval $[a,b]$. Let $x, y:[a,b]\longrightarrow\mathbb{R}$ be continuous and define
$$d(x,y)=\max_{t\in [a,b]}|x(t)-y(t)|.$$
Then $d$ is a metric on $X$. The metric space $(X.d)$ is denoted by $\mathcal{C}[a,b]$.

In a metric space $(X,d)$, nearness or closeness can be described by a neighbourhood called an $\epsilon$-ball ($\epsilon>0$) centered at $x\in X$
$$B(x,\epsilon)=\{y\in X: d(x,y)<\epsilon\}.$$
These $\epsilon$-balls form a base for the topology on $X$, called the topology on $X$ induced by the metric $d$.

Next time, we will discuss two more examples of metric spaces $\ell^p$ and $L^p$. These examples are particularly important in functional analysis as they become Banach spaces. In particular, they become Hilbert spaces when $p=2$.

# Group Theory 1: An Overview

This is the first of a series of lecture notes on group theory I intend to write for undergraduate Modern Algebra I course I am teaching in the fall semester. Before we begin to discuss the subject, I would like to give an overview of what we study in group theory or more generally in algebra.

Algebra (as a subject) is the study of algebraic structures. So, what is an algebraic structure? An algebraic structure or an algebra in short $\underline{A}$ is a non-empty set $A$ with a binary operation $f$. $\underline{A}$ is usually written as the ordered pair
$$\underline{A}=(A,f).$$
A binary operation $f$ on a set $A$ is a function $f: A\times A\longrightarrow A$. An example of a binary operation is addition $+$ on the set of integers $\mathbb{Z}$. $+$ is a function $+:\mathbb{Z}\times\mathbb{Z}\longrightarrow\mathbb{Z}$ defined by $+(1,1)=2$, $+(1,2)=3$, and so on. We usually write $+(1,1)=2$ as $1+1=2$. In general, one may consider an $n$-ary operation $f:\prod_{i=1}^n A\longrightarrow A$, where $\prod_{i=1}^n A$ denotes the $n$-copies of $A$, $A\times A\times\cdots\times A$.

There are many different kinds of algebras. Let me mention some of algebras with a binary operation here. For starter, $(A,\cdot)$, a non-empty set $A$ with a binary operation $\cdot$ is called a groupoid. A groupoid $(A,\cdot)$ with associative law
$$(ab)c=a(bc)$$
for any $a,b,c\in A$ is callaed a semigroup. If the semigroup has an identity element $e\in A$ i.e.
$$ae=ea=a$$
for any $a\in A$, it is called a monoid. If for every element $a$ of the monoid $A$, there exists an inverse element $a^{-1}\in A$ such that $aa^{-1}=a^{-1}a=e$, the monoid is called a group. A group $(A,\cdot)$ with commutative law i.e.
$$ab=ba,$$
for any $a,b\in A$ is called an abelian group named after a Norwegian mathematician Niels Abel. Note the inverse ${}^{-1}$ can be regarded as an operation on $A$, a unary operation ${}^{-1}: A\longrightarrow A$ defined by ${}^{-1}(a)=a^{-1}$ for each $a\in A$. The identity element $e$ can be also regarded as an operation, a nullary operation $e:\{\varnothing\}\longrightarrow A$. Thus, formally a group can be written as $(A,\cdot,{}^{-1},e)$, a quadrupple of a nonempty set, a binary operation, a unary operation, and a nullary operation.

Now we know what a group is and apparently, group theory is the study of groups. But what exactly are we studying there? What I am about to say is not really limited to group theory but commonly applies to studying other algebraic structures as well. There are briefly two main objectives with studying groups. One is the classification of groups. This becomes particularly interesting with groups of finite order. Here the order of a group means the number of elements of a group. We would like to answer the question “how many different groups of order $n$ are there for each $n$ and what are they?” The classification gets harder as $n$ gets larger. There are groups with the same order that appear to be different. But don’t be decieved by the appearance. They may actually be the same group. What do we mean by same here? We say two groups of the same order same if there is a one-to-one and onto map (a bijection) that preserves operations. Such a map is called an isomorphism. It turns out that if a map $\psi: G\longrightarrow G’$ from a group $G$ to another group $G’$ preserves binary operation, it automatically preserves unary and nullary operations. Here we mean preserving binary operation by
$$\psi(ab)=\psi(a)\psi(b)$$
for any $a,b\in G$. If you have taken linear algebra (and I believe you have), you would notice that a linear map is a map that preserves vector addition and scalar multiplication. A map $\psi: G\longrightarrow G’$ which preserves binary operation is called a homomorphism. If a homomorphism $\psi: G\longrightarrow G’$ is one-to-one and onto, it is an isomorphism. An isomorphism $\psi: G\longrightarrow G$ from a group $G$ onto itself is called an automorphism. In group theory, if there is an isomorphism from a group to another group, we do not distinguish them no matter how different they appear to look. The other objective is to discover new groups from old groups. Some of the new groups may be smaller in size than the old ones. Here we mean smaller in size by having a smaller number of elements i.e. having a lesser order. Some examples are subgroups and quotient groups. Some of the new groups are larger in size than the old ones. An example is direct products. Subgroups, quotient groups (also called factor groups), direct products are the things we will study as means to get new groups from old groups.

Group theory has a significance in geometry. In geometry, symmetry plays an important role. There are different types of symmetries: reflections, rotations, and translations. An interesting connection between geometry and group theory is that these symmetries form groups (symmetry groups). The most general symmetry group of finite order is called a symmetric group. In mathematics, the embedding theorem is conceptually and philosophically important (though it may be practically less important). When we study mathematics, we often feel that the structures we study are highly abstract and we feel like they only exist in our consciousness but not in the physical world. The embedding theorem tells that those abstract structures we study are indeed substructures of a larger structure that we are familiar with in the physical world. The embedding theorem implicates that we are not making up those abstract mathematical structures but we are merely discovering them which already exist in the universe. This kind of view point is called Mathematical Platonism. It turns out that there is an embedding theorem in finite group theory, namely every group of finite order is a subgroup of a symmetric group. The embedding theorem is called Cayley theorem. This means that the study of finite groups boils down to studying symmetric groups.

Remark. There is a mathematical structure called algebras over field $K$ (usually $K=\mathbb{R}$ or $K=\mathbb{C}$). An algebra $\mathcal{A}$ over field $K$ is a vector space over $K$ with a product $\cdot:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ which is distributive over addition:
$$a(b+c)=ab+ac,\ (a+b)c=ac+bc,\ \forall\ a,b,c\in\mathcal{A}.$$
(Here, the symbol $\forall$ is a logical symbol which has meaning “for each”, “for any”, “for every”, or “for all” depending on the context. I will talk more about logical symbols next time as I will use them often.) Note that an algebra $\mathcal{A}$ over field $K$ is not an algebra because the scalar product is not an operation on $\mathcal{A}$. The scalar product is in fact an action of the multiplicative group $K\setminus\{0\}$ on $\mathcal{A}$. Algebras over field $K$ are important structures in functional analysis.