Chapter 4.5, Problem 1E

Let $G$ be the group and $H$ the subgroup given in each of the following exercises of Section 4.4. In each case, is $H$ normal in $G$?

Exercise 3 b. Exercise 4 c. Exercise 5

d. Exercise 6 e. Exercise 7 f. Exercise 8

Section 4.4

Let $H$ be the subgroup $\{e,\beta \}$ of the octic group $D_4$.

Find the distinct left cosets of $H$ in $D_4$, write out their elements, partition $D_4$ into left cosets of $H$, and give $[D_4:H]$.

Find the distinct right cosets of $H$ in $D_4$, write out their elements, and partition $D_4$ into right cosets of $H$.

Let $H$ be the subgroup $\{e,\Delta \}$ of the octic group $D_4$.

Find the distinct left cosets of $H$ in $D_4$, write out their elements, partition $D_4$ into left cosets of $H$, and give $[D_4:H]$.

Find the distinct right cosets of $H$ in $D_4$, write out their elements, and partition $D_4$ into right cosets of $H$.

Let $H$ be the subgroup $\{e,\Delta \}$ of the octic group $D_4$.

Find the distinct left cosets of $H$ in $D_4$, write out their elements, partition $D_4$ into left cosets of $H$, and give $[D_4:H]$.

Find the distinct right cosets of $H$ in $D_4$, write out their elements, and partition $D_4$ into right cosets of $H$.

Let $H$ be the subgroup $\{(1),(2,3)\}$ of $S_3$.

Find the distinct left cosets of $H$ in $S_3$, write out their elements, partition $S_3$ into left cosets of $H$, and give $[S_3:H]$.

Find the distinct right cosets of $H$ in $S_3$, write out their elements, and partition $S_3$ into right cosets of $H$.

In Exercises 7 and 8, let $G$ be the multiplicative group of permutation matrices $\{I_3,P_3,P_3{}^2,P_1,P_4,P_2\}$ in Example 6 of Section 3.5

Let $H$ be the subgroup of $G$ given by $H=\{I_3,P_4\}=\left\{\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\right\}$.

Find the distinct left cosets of $H$ in $G$, write out their elements, partition $G$ into left cosets of $H$, and give $[G:H]$.

Find the distinct right cosets of $H$ in $G$, write out their elements, and partition $G$ into right cosets of $H$.

Let $H$ be the subgroup of $G$ given by $H=\{I_3,P_3,P_3^2\}=\left\{\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)\right\}$.

Find the distinct left cosets of $H$ in $G$, write out their elements, partition $G$ into left cosets of $H$, and give $[G:H]$.

Find the distinct right cosets of $H$ in $G$, write out their elements, and partition $G$ into right cosets of $H$.