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## Reading Questions10.3Reading Questions

###### 1.

Let $G$ be the group of symmetries of an equilateral triangle, expressed as permutations of the vertices numbered $1,2,3\text{.}$ Let $H$ be the subgroup $H=\langle (1\,2) \rangle\text{.}$ Build the left and right cosets of $H$ in $G\text{.}$

###### 2.

Based on your answer to the previous question, is $H$ normal in $G\text{?}$ Explain why or why not.

###### 3.

The subgroup $8\mathbb Z$ is normal in $\mathbb Z\text{.}$ In the factor group $\mathbb Z/8\mathbb Z$ perform the computation $(3+8\mathbb Z)+(7+8\mathbb Z)\text{.}$

###### 4.

List two statements about a group $G$ and a subgroup $H$ that are equivalent to “$H$ is normal in $G\text{.}$”

###### 5.

In your own words, what is a factor group?