## Section2.3Subgroups and cosets

### Subsection2.3.1

###### 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 $G\text{.}$ A (left) coset of a subgroup $H$ of $G$ is a set of the form

\begin{equation*} gH := \{gh \colon h\in H\}. \end{equation*}

The set of all cosets of $H$ is denoted $G/H\text{.}$

\begin{equation*} G/H := \{gH\colon g\in G\} \end{equation*}

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.

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 $G=\langle g\rangle\text{.}$

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

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. $\displaystyle \langle 6,8\rangle =\langle 2\rangle = 2\Z$

### Exercises2.3.2Exercises

###### 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*}

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. Prove Proposition 2.3.9.
2. Now suppose that a group $G$ is finite. Show that all of the cosets of a subgroup $H$ have the same size.
3. 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

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.