Question

Find (a) all the proper subgroups and (b) all the conjugacy classes of the symmetry group of a regular pentagon.

Short answer

Proper subgroups: \( \{e\} \), \( \{e, r^2\} \), and each reflection subgroup. Conjugacy classes: \( \{e\} \), \( \{r, r^4\} \), \( \{r^2, r^3\} \), and \( \{s_1, s_2, s_3, s_4, s_5\} \).

Step-by-step solution

- Understand the symmetry group

The symmetry group of a regular pentagon is the Dihedral group of order 10, denoted as \( D_5 \). It consists of 5 rotations and 5 reflections.

- Determine the proper subgroups

Identify subgroups that are proper (i.e., non-trivial and not the whole group). Subgroups to consider are rotation subgroups and reflection subgroups. The subgroup of rotations: \( \text{Rotations} = \{ e, r, r^2, r^3, r^4 \} \) where \( r \) is the rotation by \( \frac{2\text{π}}{5} \). The subgroup of order 2: \( \{ e, r^2 \} \). Reflection subgroups: Each symmetry by reflection across the axis passing through a vertex and the midpoint of the opposite side forms a subgroup of order 2.

- List all proper subgroups

Therefore, the proper subgroups of \( D_5 \) are \( \{e\} \), \( \{ e, r^2 \} \), and each reflection subgroup, such as \( \{ e, s_1 \} \), \( \{ e, s_2 \} \), \( \{ e, s_3 \} \), \( \{ e, s_4 \} \), and \( \{ e, s_5 \} \).

- Determine the conjugacy classes

Conjugacy classes in \( D_5 \) group the elements that are conjugate to each other. For rotations, the classes are \( \{ e \} \), \( \{ r, r^4 \} \), and \( \{ r^2, r^3 \} \) because rotations and their inverses are conjugate. For reflections, since each reflection axis can only be interchanged with other reflections: the classes are \( \{ s_1, s_2, s_3, s_4, s_5 \} \).

- List all conjugacy classes

The conjugacy classes are \( \{ e \} \), \( \{ r, r^4 \} \), \{ r^2, r^3 \} \}, and \( \{ s_1, s_2, s_3, s_4, s_5 \} \).

Key concepts

Proper Subgroups

A proper subgroup is a subset of a group that is not trivial (contains more than just the identity element) and is not the entire group itself. For the dihedral group, denoted as \(D_5\), which represents the symmetries of a regular pentagon, there are both rotational and reflection subgroups.
The subgroup of rotations includes all the rotations that map the pentagon onto itself: \( \text{Rotations} = \{ e, r, r^2, r^3, r^4 \} \), where \(r\) represents a rotation by \(\frac{2\pi}{5}\) radians. Proper subgroups of this include \( \{ e, r^2 \} \), where the subgroup is of order 2.
Each reflection across an axis forms its own subgroup of order 2. For example, the reflections could be \( \{ e, s_1 \} \), \( \{ e, s_2 \} \), and so on up to \( \{ e, s_5 \} \). Thus, the proper subgroups of \( D_5 \) are all sets that include the identity and one of these specific symmetries.

Conjugacy Classes

Conjugacy classes help in understanding the structure of a group by grouping elements that can be transformed into each other by other elements in the group. In \(D_5\), elements that are conjugate will lie in the same class. This classification helps in simplifying group analysis.
The identity element \(e\) in \(D_5\) stands alone in its own conjugacy class: \( \{ e \} \).
For rotations, since a rotation and its inverse (the reverse rotation) can replace each other, the conjugacy classes include pairs. Thus, \( \{ r, r^4 \} \) and \( \{ r^2, r^3 \} \) are such classes.
Reflections, on the other hand, can only be interchanged among themselves. Therefore, all reflections form a single conjugacy class: \( \{ s_1, s_2, s_3, s_4, s_5 \} \).

Symmetry Group

The symmetry group of a geometric object consists of all transformations that leave the object unchanged. For a regular pentagon, these transformations include rotations and reflections.
The set of these transformations forms the Dihedral group \(D_5\), with an order of 10, since there are 5 rotations and 5 reflections. Each operation in the symmetry group maintains the geometric properties of the pentagon.
The elements of the symmetry group of the pentagon include:

• The identity transformation (doing nothing)
• Five rotations (including the identity rotation)
• Five reflections

Totaling these transformations gives us the full symmetry group \(D_5\). Thus, the symmetry group plays a crucial role in studying the properties and accessible symmetries of the pentagon.

Rotations

Rotations in the context of the dihedral group of a pentagon refer to turning the pentagon about its center. Each rotation maps the pentagon onto itself.
The dihedral group \(D_5\) features five rotations, each by an angle of \( \frac{2k\pi}{5} \) where \(k\) ranges from 0 to 4. These rotations can be listed as:

• \(e\) – the identity rotation (0 degrees)
• \(r\) – rotation by \(\frac{2\pi}{5}\) (72 degrees)
• \(r^2\) – rotation by \(\frac{4\pi}{5}\) (144 degrees)
• \(r^3\) – rotation by \(\frac{6\pi}{5}\) (216 degrees)
• \(r^4\) – rotation by \(\frac{8\pi}{5}\) (288 degrees)

Each of these rotations reintegrates the pentagon back onto its original position, maintaining the shape but altering the points' positions.

Reflections

Reflections in the dihedral group represent flips over axes that intersect the pentagon at specific points. For a regular pentagon, each reflection axis goes through one vertex and the midpoint of the opposite side.
The reflection axes for \( D_5 \) can be labeled as \( s_1, s_2, s_3, s_4, s_5 \), each bisecting the pentagon symmetrically. Each of these reflections swaps points symmetrically around the axis.
For example, reflection \( s_1 \) maps a vertex to its mirror image across the axis running through the opposite side's midpoint. Similar transformations happen with \( s_2 \), \( s_3, \) etc.
Each reflection operation within \(D_5\) keeps the pentagon's overall structure intact but changes the arrangement of vertices. Therefore, these reflections are essential to understanding the full symmetry operations of the pentagon.