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# Section7.2Introduction to cosets and normal subgroups

##### Definition7.2.1

Given a group $G$ with subgroup $H\text{,}$ we define $\siml$ on $G$ by

\begin{equation*} a\siml b \text{ if and only if } a^{-1}b\in H \end{equation*}

and $\simr$ on $G$ by

\begin{equation*} a\simr b \text{ if and only if } ab^{-1}\in H. \end{equation*}
##### Remark7.2.3

Of course, different subgroups $H$ and $K$ in a group $G$ will give rise to different relations $\siml$ and $\simr$ on $G\text{;}$ that is, these relations are really defined with respect to a particular subgroup of $G\text{.}$

From now on, whenever we discuss $\siml$ or $\simr$ on a group, assume that it is with respect to a particular subgroup $H$ of $G\text{.}$

Now, as equivalence relations on a group $G\text{,}$ each of $\siml$ and $\simr$ gives rise to a partition of $G\text{.}$ What are the cells of those partitions?

##### Definition7.2.4

Given $a\in G\text{,}$ we define

\begin{equation*} aH = \{ah\,:\, h\in H\} \end{equation*}

and

\begin{equation*} Ha=\{ha\,:\,h\in H\}. \end{equation*}

We call $aH$ and $Ha\text{,}$ respectively, the left and right cosets of $H$ containing $a$.

If we know that $G$ is abelian, with operation denoted by $+\text{,}$ we may denote these left and right cosets by $a+H$ and $H+a\text{,}$ respectively.

##### Note7.2.5

In the following, we use the notation $\Leftrightarrow$ to denote the phrase “if and only if.”

##### Proof

We next summarize some facts about the left and right cosets of a subgroup $H$ of a group $G\text{:}$

##### Remark7.2.8

We can use Statements 2 and 3, above, to save some time when computing left and right cosets of a subgroup of a group.

##### Example7.2.9

Find the left and right cosets of $H=\langle (12)\rangle$ in $S_3\text{.}$

The left cosets are

\begin{equation*} eH=H=(12)H, \end{equation*} \begin{equation*} (13)H=\{(13),(123)\}=(123)H, \end{equation*} \begin{equation*} \text{ and } (23)H=\{(23),(132)\}=(132)H, \end{equation*}

and the right cosets are

\begin{equation*} He=H=H(12), \end{equation*} \begin{equation*} H(13)=\{(13),(132)\}=H(132), \end{equation*} \begin{equation*} \text{ and } H(23)=\{(23),(123)\}=H(123). \end{equation*}

Thus, $\siml$ partitions $S_3$ into $\{H,\{(13),(123)\},\{(23), (132)\}\}$ and $\simr$ partitions $S_3$ into $\{H,\{(13),(132)\},\{(23), (123)\}\}\text{.}$

##### Example7.2.10

Find the left and right cosets of $H=\langle f\rangle$ in $D_4\text{.}$

This example is left as an exercise for the reader.

We now draw attention to some very important facts:

##### Warning7.2.11

For $a,b\in G\text{:}$

1. In general, $aH \neq Ha\text{!}$

2. $aH=bH$ does not necessarily imply $a=b$ or that there exists an $h\in H$ with $ah=bh\text{;}$ similarly, $Ha=Hb$ does not necessarily imply $a=b$ or that there exists an $h\in H$ with $ha=hb\text{.}$

##### Example7.2.12

We saw above that in $S_3$ with $H=\langle (12)\rangle\text{,}$

\begin{equation*} (13)H=\{(13),(123)\} \neq \{(13),(132)\}=H(13). \end{equation*}

Also, $(13)H=(123)H$ but $(13)e\neq (123)e$ and $(13)(12)\neq (123)(12)\text{.}$

It turns out that subgroups $H$ for which $aH=Ha$ for all $a\in G$ will be very important to us.

##### Definition7.2.13

We say that subgroup $H$ of $G$ is normal in $G$ (or is normal subgroup of $G$) if $aH=Ha$ for all $a\in G\text{.}$ We denote that fact that $H$ is normal in $G$ by writing $H\unlhd G\text{.}$

##### Remark7.2.14

1. If $H$ is normal in $G\text{,}$ we may refer to the left and right cosets of $G$ as simply cosets.

2. Of course, if $G$ is abelian, every subgroup of $G$ is normal in $G\text{.}$ But there can also be normal subgroups of nonabelian groups: for instance, the trivial and improper subgroups of every group are normal in that group.

##### Example7.2.15

Find the cosets of $5\Z$ in $\Z\text{.}$

Notice that in additive notation, the statement “$a^{-1}b\in H$” becomes $-a+b\in H\text{.}$ So for $a,b\in \Z\text{,}$ $a\siml b$ if and only if $-a+b \in 5\Z\text{;}$ that is, if and only if $5$ divides $b-a\text{.}$ In other words, $a\siml b$ if and only if $a\equiv_5 b\text{.}$ So in this case, $\siml$ is just congruence modulo $5\text{.}$ Thus, the cosets of $5\Z$ in $\Z$ are

\begin{align*} 5\Z\amp =\{\ldots,-5,0,5,\ldots\}\\ 1+5\Z\amp =\{\ldots,-4, 1, 6,\ldots\},\\ 2+5\Z\amp =\{\ldots,-3,2, 7, \ldots\},\\ 3+5\Z\amp =\{\ldots,-2,3, 8, \ldots\},\\ 4+5\Z\amp =\{\ldots,-1, 4, 9, \ldots\}. \end{align*}

Do you see how this example would generalize for $n\Z$ ($n \in \Z^+$) in $\Z\text{?}$

##### Example7.2.16

Find the cosets of $H=\langle 12\rangle$ in $4\Z\text{.}$

They are

\begin{align*} H\amp =\{\ldots, -12,0,12\ldots\},\\ 4+H \amp = \{\ldots,-8,4,16,\ldots\},\\ 8+H\amp =\{\ldots, -4,8,20,\ldots\}. \end{align*}
##### Example7.2.17

Find the cosets of $H=\langle 4\rangle$ in $\Z_{12}\text{.}$

They are

\begin{align*} H\amp =\{0,4,8\},\\ 1+H \amp = \{1,5,9\},\\ 2+H\amp =\{2,6,10\}\\ 3+H\amp =\{3,7,11\}. \end{align*}

We now consider the set of all left cosets of a subgroup of a group.

##### Definition7.2.18

We let $G/H$ be the set of all left cosets of subgroup $H$ in $G\text{.}$ We read $G/H$ as “$G$ mod $H\text{.}$”

(We may denote the set of all right cosets of subgroup $H$ in $G$ by $H\backslash G\text{,}$ but we will not use that notation in this class.)