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Reading Questions 10.3 Reading 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?

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