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# 数学代写|抽象代数代写Abstract Algebra代考|MATH411 Right cosets

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## 数学代写|抽象代数代写Abstract Algebra代考|Right cosets

In Section 19.1, we considered left cosets of the form $a H$, where we multiplied each element of $H$ on the left by $a$. We can also consider right cosets $H a$, as shown in the example below.

Example 19.7. Consider again the group $U_{13}$ and its subgroup $H={1,3,9}$. The left coset $6 H$ is given by $6 H={6 \cdot 1,6 \cdot 3,6 \cdot 9}={6,5,2}$, and we have the right coset $H 6={1 \cdot 6,3 \cdot 6,9 \cdot 6}={6,5,2}$. Observe that $6 H=H 6$; i.e., the left and right cosets are equal, because $U_{13}$ is commutative.

As seen in Example 19.7, the distinction between left and right cosets is irrelevant in a commutative group. In particular, additive groups are always commutative, so there is no distinction between the left coset $a+H$ and the right coset $H+a$. Thus, we will only consider left cosets $a+H$ with additive groups.
Here is a non-commutative example, where things get a bit more interesting.
Example 19.8. Let $H={\varepsilon, d}$ be a subgroup of $D_4$. Let’s compute and compare the left coset $r_{90} \mathrm{H}$ and the right coset $H r_{90}$ :

• $r_{90} H=\left{r_{90} \cdot \varepsilon, r_{90} \cdot d\right}=\left{r_{90}, h\right}$.
• $H r_{90}=\left{\varepsilon \cdot r_{90}, d \cdot r_{90}\right}=\left{r_{90}, v\right}$.
Therefore, the left and right cosets are not the same; i.e., $r_{90} H \neq H r_{90}$.
Example 19.9. Let $K=C(h)=\left{\varepsilon, r_{180}, h, v\right}$ be a subgroup of $D_4$. (It’s the centralizer of $h$ in $D_4$. See Section 5.3.) Let’s compute and compare the left coset $d K$ and the right $\operatorname{coset} K d$ :
• $d K=\left{d \cdot \varepsilon, d \cdot r_{180}, d \cdot h, d \cdot v\right}=\left{d, d^{\prime}, r_{270}, r_{90}\right}$.
• $K d=\left{\varepsilon \cdot d, r_{180} \cdot d, h \cdot d, v \cdot d\right}=\left{d, d^{\prime}, r_{90}, r_{270}\right}$
Thus we have a coset equality $d K=K d$, because these sets contain the same four elements. But this does not imply that we have an element-by-element equality; i.e., $d k=k d$ for all $k \in K$. Indeed, we have $d h \neq h d$ and $d v \neq v d$, where $h, v \in K$.

## 数学代写|抽象代数代写Abstract Algebra代考|Properties of cosets

We’ve seen plenty of examples of cosets thus far, and now we’re ready for a general definition.
Definition $19.10$ (Coset). Let $G$ be a group, $H$ a subgroup of $G$, and $a \in G$. Then:

• The set $a H={a h \mid h \in H}$ is the left coset of $H$ generated by $a$.
• The set $H a={h a \mid h \in H}$ is the right coset of $H$ generated by $a$.
The element $a$ is called the coset representative of $a H$ and $H a$.
Remark. If $G$ is an additive group, then the left and right cosets are $a+H={a+h \mid$ $h \in H}$ and $H+a={h+a \mid h \in H}$, respectively. Recall that we always have $a+H=H+a$, since additive groups are commutative. Given this lack of distinction between left and right cosets, we will only consider left cosets $a+H$ with additive groups.

Below are the first three properties of cosets observed in Examples 19.1 and 19.5. (The fourth property about how the distinct cosets partition the group will be addressed in the next chapter.) While they are stated in the context of left cosets, as will be typical of coset theorems, analogous statements are true for right cosets. Each proof is written for a multiplicative group, and the proofs for an additive group are left for you as an exercise.
The following example motivates the proof of the first theorem.

## 数学代写|抽象代数代写Abstract Algebra代考|Right cosets

〈left 的分隔符缺失或无法识别

〈left 的分隔符缺失或无法识别

〈left 的分隔符缺失或无法识别

lleft 的分隔符缺失或无法识别

$\mathrm{IE} ， d k=k d$ 对所有人 $k \in K$. 确实，我们有 $d h \neq h d$ 和 $d v \neq v d$ ，在哪里 $h, v \in K$.

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