10.3: Reading Questions
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 \(G\text{.}\)
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?