Abstract Algebra: Theory and Applications

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 $H=⟨(12)⟩\text{.}$ Build the left and right cosets of $H$ in $G\text{.}$

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

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

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

In your own words, what is a factor group?