36. Compute the subgroups of the symmetry group of a square.

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To compute the subgroups of the symmetry group of a square, we first need to identify the symmetry group itself. The symmetry group of a square, denoted as $$ D_4 $$, consists of all the rigid motions (isometries) that map the square onto itself. This group includes rotations and reflections.

### Elements of $$ D_4 $$

The symmetry group $$ D_4 $$ has the following elements:

1. **Rotations:**
   - $$ e $$: Identity (0 degrees)
   - $$ r $$: Rotation by 90 degrees
   - $$ r^2 $$: Rotation by 180 degrees
   - $$ r^3 $$: Rotation by 270 degrees

2. **Reflections:**
   - $$ s_1 $$: Reflection across the vertical axis
   - $$ s_2 $$: Reflection across the horizontal axis
   - $$ s_3 $$: Reflection across the diagonal from top-left to bottom-right
   - $$ s_4 $$: Reflection across the diagonal from top-right to bottom-left

### Order of $$ D_4 $$

The order of the group $$ D_4 $$ is 8, as it contains 8 elements.

### Subgroups of $$ D_4 $$

To find the subgroups, we can use Lagrange's theorem, which states that the order of a subgroup must divide the order of the group. Therefore, the possible orders of subgroups of $$ D_4 $$ are 1, 2, 4, and 8.

1. **Trivial subgroup:**
   - $$ \{ e \} $$ (order 1)

2. **Subgroups of order 2:**
   - $$ \{ e, r^2 \} $$ (rotation by 180 degrees)
   - $$ \{ e, s_1 \} $$ (reflection across the vertical axis)
   - $$ \{ e, s_2 \} $$ (reflection across the horizontal axis)
   - $$ \{ e, s_3 \} $$ (reflection across the diagonal from top-left to bottom-right)
   - $$ \{ e, s_4 \} $$ (reflection across the diagonal from top-right to bottom-left)

3. **Subgroups of order 4:**
   - $$ \{ e, r^2, r, r^3 \} $$ (the subgroup generated by rotations)
   - $$ \{ e, s_1, s_2, r^2 \} $$ (reflections across the axes and the 180-degree rotation)
   - $$ \{ e, s_3, s_4, r^2 \} $$ (reflections across the diagonals and the 180-degree rotation)

4. **The whole group:**
   - $$ D_4 $$ itself (order 8)

### Summary of Subgroups

The subgroups of the symmetry group $$ D_4 $$ are:

1. $$ \{ e \} $$ (order 1)
2. $$ \{ e, r^2 \} $$ (order 2)
3. $$ \{ e, s_1 \} $$ (order 2)
4. $$ \{ e, s_2 \} $$ (order 2)
5. $$ \{ e, s_3 \} $$ (order 2)
6. $$ \{ e, s_4 \} $$ (order 2)
7. $$ \{ e, r^2, r, r^3 \} $$ (order 4)
8. $$ \{ e, s_1, s_2, r^2 \} $$ (order 4)
9. $$ \{ e, s_3, s_4, r^2 \} $$ (order 4)
10. $$ D_4 $$ (order 8)

These subgroups represent all the possible combinations of rotations and reflections that can be formed from the symmetry group of a square.
The symmetry group of a square, $$ D_4 $$, has 8 elements consisting of rotations and reflections. Its subgroups include: 1. The trivial group with just the identity. 2. Four subgroups of order 2, each containing the identity and one reflection. 3. Three subgroups of order 4, each containing the identity, a 180-degree rotation, and two reflections. 4. The entire group $$ D_4 $$ itself. These subgroups represent all possible combinations of rotations and reflections that map the square onto itself.

36. Compute the subgroups of the symmetry group of a square.

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