# Introduction to Groups and Geometries

## Section2.3Subgroups and cosets

### Definition2.3.1.Subgroups and cosets.

A subset $$H$$ of a group $$G$$ is called a subgroup of $$G$$ if $$H$$ itself is a group under the group operation of $$G$$ restricted to $$H\text{.}$$ We write $$H\leq G$$ to indicate that $$H$$ is a subgroup of $$G\text{.}$$ Given a subgroup $$H$$ of $$G\text{,}$$ and given an element $$g$$ in $$G\text{,}$$ the set
\begin{equation*} gH := \{gh \colon h\in H\} \end{equation*}
is called a (left) coset of $$H\text{.}$$ The set of all cosets of $$H$$ is denoted $$G/H\text{.}$$
\begin{equation*} G/H := \{gH\colon g\in G\} \end{equation*}

### Checkpoint2.3.2.

Consider $$D_4$$ as described in Checkpoint 2.1.6.
\begin{equation*} D_4=\{R_0,R_{1/4},R_{1/2},R_{3/4},F_H,F_V,F_D,F_{D'}\} \end{equation*}
1. Is the subset $$\{R_0,R_{1/4},R_{1/2},R_{3/4}\}$$ of rotations a subgroup of $$D_4\text{?}$$ Why or why not?
2. Is the subset $$\{F_H,F_V,F_D,F_{D'}\}$$ of reflections a subgroup of $$D_4\text{?}$$ Why or why not?
1. Yes. The composition of any two rotations is a rotation, and every rotation has an inverse that is also a rotation.
2. No. Just observe that $$F_H^2=R_0$$ is not a reflection. The group operation on $$D_4$$ does not restrict properly to the subset of reflections.

### Checkpoint2.3.3.

Find $$G/H$$ for $$G=S_3\text{,}$$ $$H=\{e,(12)\}\text{.}$$
\begin{align*} G/H \amp =\{eH, (12)H, (13)H, (23)H, (123)H, (132)H\}\\ \amp = \{\{e,(12)\},\{(12),e\},\{(13),(123)\},\{(23),(132)\},\{(13),(123)\},\{(132),(23)\}\}\\ \amp = \{H,\{(13),(123)\},\{(23),(132)\}\} \end{align*}
Comment on notational convention: If $$S=\{s_1,s_2,\ldots,s_k\}$$ is finite, we write $$\langle s_1,s_2,\ldots,s_k\rangle$$ for $$\langle S\rangle\text{,}$$ instead of the more cumbersome $$\langle \{s_1,s_2,\ldots,s_k\}\rangle\text{.}$$

### Observation2.3.6.

If $$G$$ is a cyclic group with generator $$g\text{,}$$ then $$G=\langle g\rangle\text{.}$$

### Checkpoint2.3.7.

Show that $$\langle S\rangle$$ is indeed a subgroup of $$G\text{.}$$ How would this fail if $$S$$ were empty?

### Checkpoint2.3.8.

1. Find $$\langle F_H,F_V\rangle\subseteq D_4\text{.}$$
2. Find $$\langle 6,8\rangle \subseteq \Z\text{.}$$
1. $$\displaystyle \langle F_H,F_V\rangle=\{R_0,R_{1/2},F_H,F_V\}$$
2. $$\displaystyle \langle 6,8\rangle =\langle 2\rangle = 2\Z$$

### ExercisesExercises

#### 2.

Find all the subgroups of $$S_3\text{.}$$
In the "list of values" permutation notation of Checkpoint 2.1.2, the subgroups of $$S_3$$ are $$\{[1,2,3]\}\text{,}$$ $$\{[1,2,3],[2,1,3]\}\text{,}$$ $$\{[1,2,3],[1,3,2]\}\text{,}$$ $$\{[1,2,3],[3,2,1]\}\text{,}$$ $$\{[1,2,3],[2,3,1],[3,1,2]\}\text{,}$$ and $$S_3\text{.}$$ In cycle notation, the subgroups of $$S_3$$ (in the same order) are $$\{e\}\text{,}$$ $$\{e,(12)\}\text{,}$$ $$\{e,(23)\}\text{,}$$ $$\{e,(13)\}\text{,}$$ $$\{e,(123),(132)\}\text{,}$$ $$S_3\text{.}$$

#### 3.

Find all the cosets of the subgroup $$\{R_0,R_{1/2}\}$$ of $$D_4\text{.}$$

#### 4.Subgroups of $$\Z$$ and $$\Z_n$$.

1. Let $$H$$ be a subgroup of $$\Z\text{.}$$ Show that either $$H=\{0\}$$ or $$H=\langle d\rangle\text{,}$$ where $$d$$ is the smallest positive element in $$H\text{.}$$
2. Let $$H$$ be a subgroup of $$\Z_n\text{.}$$ Show that either $$H=\{0\}$$ or $$H=\langle d\rangle\text{,}$$ where $$d$$ is the smallest positive element in $$H\text{.}$$
3. Let $$n_1,n_2,\ldots,n_r$$ be positive integers. Show that
\begin{equation*} \langle n_1,n_2,\ldots,n_r \rangle = \langle \gcd(n_1,n_2,\ldots,n_r)\rangle. \end{equation*}
Hint.
Suggestion: Do the case $$r=2$$ first.
Consequence of this exercise: The greatest common divisor $$\gcd(a,b)$$ of integers $$a,b$$ is the smallest positive integer of the form $$sa+tb$$ over all integers $$s,t\text{.}$$ Two integers $$a,b$$ are relatively prime if and only if there exist integers $$s,t$$ such that $$sa+tb=1\text{.}$$

#### 5.Centralizers, Center of a group.

The centralizer of an element $$a$$ in a group $$G\text{,}$$ denoted $$C(a)\text{,}$$ is the set
\begin{equation*} C(a) = \{g\in G\colon ag=ga\}. \end{equation*}
The center of a group $$G\text{,}$$ denoted $$Z(G)\text{,}$$ is the set
\begin{equation*} Z(G) = \{g\in G\colon ag=ga \;\; \forall a\in G\}. \end{equation*}
1. Show that the centralizer $$C(a)$$ of any element $$a$$ in a group $$G$$ is a subgroup of $$G\text{.}$$
2. Show that the center $$Z(G)$$ of a group $$G$$ is a subgroup of $$G\text{.}$$

#### 6.The order of a group element.

Let $$g$$ be an element of a group $$G\text{.}$$ The order of $$g\text{,}$$ denoted $$|g|\text{,}$$ is the smallest positive integer $$n$$ such that $$g^n=e\text{,}$$ if such an integer exists. If there is no positive integer $$n$$ such that $$g^n=e\text{,}$$ then $$g$$ is said to have infinite order. Show that, if the order of $$g$$ is finite, say $$|g|=n\text{,}$$ then
\begin{equation*} \langle g \rangle = \{g^0,g^1,g^2,\ldots,g^{n-1}\}\text{.} \end{equation*}
Consequence of this exercise: If $$G$$ is cyclic with generator $$g\text{,}$$ then $$|G|=|g|\text{.}$$

#### 7.Cosets of a subgroup partition the group, Lagrange’s Theorem.

1. Now suppose that a group $$G$$ is finite. Show that all of the cosets of a subgroup $$H$$ have the same size.
2. Prove the following.
##### Lagrange’s Theorem.
If $$G$$ is a finite group and $$H$$ is a subgroup, then the order of $$H$$ divides the order of $$G\text{.}$$
Hint.
For part (b), let $$aH,bH$$ be cosets. Show that the function $$aH\to bH$$ given by $$x\to ba^{-1}x$$ is a bijection.

#### 8.Consequences of Lagrange’s Theorem.

1. Show that the order of any element of a finite group divides the order of the group.
2. Let $$G$$ be a finite group, and let $$g\in G\text{.}$$ Show that $$g^{|G|}=e\text{.}$$
3. Show that a group of prime order is cyclic.