Question: for each design given in picture(attached) determine the symmetry group.
My attempt: For first design in first row, i think group of symmetry is $D_4$ i.e. Dihedral group of order $8$. Since first design has four rotational symmetries (rotation by $0°$, rotation by $90°$, rotation by $180°$, rotation by $270°$) and four reflection symmetries.
For second design (from left side) in first row, i think group of symmetry is cyclic rotation group of order $3$ and hence isomorphic to $\mathbb{Z}_3$. since second design has three rotational symmetries but not reflectional symmetries.
For third design in first row i am not sure.
For fourth design i think it is $D_{16}$. i.e Dihedral group of order $32$.
For first design in second row, i think its $D_7$ i.e Dihedral group of order $14$.
For second design in second row, i think the group of symmetry is $D_4$ i e Dihedral group of order $8$.
For third design, i think its $D_5$.
For last design, i think it is also $D_5$.
I am not sure of my answer and i dont have solution/key for this exercise. Please help.
That's a good attempt.
The groups, respectively, are
$$\begin{array} & & & \\ D_4 & \Bbb Z_3 & D_3 & D_{16} \\ D_7 & D_4 & D_5 & \Bbb Z_{10}, \end{array}$$
where $D_n$ is the dihedral group of order $2n$.