Question:

Identify the eight symmetry operations on a square. Show that they form a group ${D}_{4}$ (known to crystallographers as 4mm and to chemists as ${C}_{4v}$) having one element of order 1, five of order 2 and two of order 4. Find its proper subgroups and the corresponding cosets.

Step-by-step

The operation of leaving the square alone is a trivial symmetry operation, but an
important one, as it is the identity I of the group; it has order 1.
Rotations about an axis perpendicular to the plane of the square by π/4, π/2
and 3π/2 each take the square into itself. The first and last of these have to be
repeated four times to reproduce the effect of I, and so they have order 4. The
rotation by π/2 clearly has order 2.
Reflections in the two axes parallel to the sides of the square and passing through
its centre are also symmetry operations, as are reflections in the two principal
diagonals of the square; all of these reflections have order 2.
Using the notation indicated in figure 28.1, R being a rotation of π/2 about an
axis perpendicular to the square, we have: I has order 1; ${R}^{2}, {m}_{1}, {m}_{2}, {m}_{3}, {m}_{4}$ have
order $2; R, {R}^{3}$ have order 4.
The group multiplication table takes the form
Inspection of this table shows the existence of the non-trivial subgroups listed below, and tedious but straightforward evaluation of the products of selected elements of the group with all the elements of any one subgroup provides the cosets of that subgroup. The results are as follows:
subgroup {$I,R,{R}^{2}, {R}^{3}$} has cosets {$I,R,{R}^{2}, {R}^{3}$}, {${m}_{1}, {m}_{2}, {m}_{3}, {m}_{4}$};
subgroup {$I,{R}^{2}, {m}_{1}, {m}_{2}$} has cosets {$I,{R}^{2}, {m}_{1}, {m}_{2}$}, {$I,{R}^{2}, {m}_{3}, {m}_{4}$};
subgroup {$I,{R}^{2}, {m}_{3}, {m}_{4}$} has cosets {$I,{R}^{2}, {m}_{3}, {m}_{4}$}, {$I,{R}^{2}, {m}_{1}, {m}_{2}$};
subgroup {$I,{R}^{2}$} has cosets {$I,{R}^{2}$}, {$R,{R}^{3}$}, {${m}_{1}, {m}_{2}$}, {${m}_{3}, {m}_{4}$};
subgroup {$I,{m}_{1}$} has cosets {$I,{m}_{1}$}, {$R,{m}_{3}$}, {${R}^{2}, {m}_{2}$}, {${R}^{3}, {m}_{4}$};
subgroup {$I,{m}_{2}$} has cosets {$I,{m}_{2}$}, {$R,{m}_{4}$}, {${R}^{2}, {m}_{1}$}, {${R}^{3}, {m}_{3}$};
subgroup {$I,{m}_{3}$} has cosets {$I,{m}_{3}$}, {$R,{m}_{2}$}, {${R}^{2}, {m}_{4}$}, {${R}^{3}, {m}_{1}$};
subgroup {$I,{m}_{4}$} has cosets {$I,{m}_{4}$}, {$R,{m}_{1}$}, {${R}^{2}, {m}_{3}$}, {${R}^{3}, {m}_{2}$}.