Question

Identify the eight symmetry operations on a square. Show that they form a group (known to crystallographers as \(4 \mathrm{~mm}\) or to chemists as \(C_{4 c}\) ) having one element of order 1, five of order 2 and two of order \(4 .\) Find its proper subgroups and the corresponding cosets.

Short answer

The eight symmetry operations are E, R90, R180, R270, σ_v, σ_h, σ_d1, and σ_d2. These form a group with one element of order 1, five of order 2, and two of order 4.

Step-by-step solution

- Identify Symmetry Operations

A square has the following eight symmetry operations: the identity operation (E), 90-degree rotation (R90), 180-degree rotation (R180), 270-degree rotation (R270), and reflections across vertical axis (σ_v), horizontal axis (σ_h), and the two diagonals (σ_d1, σ_d2).

- Determine Orders of Elements

The identity operation (E) has order 1. The 180-degree rotation (R180) and the reflections (σ_v, σ_h, σ_d1, σ_d2) each have order 2. The 90-degree (R90) and 270-degree (R270) rotations each have order 4.

- Verify Group Properties

To show these operations form a group, verify the group properties: Closure (combining any two operations results in another operation from the set), associativity (operations follow associative law), identity element (E acts as the identity), and inverses (each element has an inverse within the group).

- Construct Group Table

Create a group table for these symmetry operations. Ensuring closure, this table should showcase all combinations of the eight operations applied in sequence, verifying their result is another operation within the group.

- Find Proper Subgroups

Identify proper subgroups by recognizing sets of symmetry operations closed under the group operation and containing the identity. Examples: Subgroups involving {E, R180, σ_v, σ_h}, {E, R90, R180, R270}.

- Find Corresponding Cosets

Using the coset decomposition, list the cosets for each subgroup. Cosets are formed by multiplying a fixed element by each element of the subgroup: for a subgroup H and element g in the group, the left coset is gH.

Key concepts

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set equipped with an operation that combines any two elements to form a third element in such a way that four conditions, called the group axioms, are satisfied. These are Closure, Associativity, Identity, and Inverses. In the context of symmetry operations on a square, we check these axioms:

• **Closure**: Combining any two symmetry operations still results in another symmetry operation.
• **Associativity**: The order in which operations are performed does not change the result.
• **Identity**: There is an identity operation (doing nothing) that leaves the square unchanged.
• **Inverses**: Each operation has an inverse operation that undoes it.

By following these, we confirm that the set of symmetry operations on a square forms a group.

Symmetry Elements

A symmetry element is a point, line, or plane about which symmetry operations are performed. For a square, the symmetry elements include:

• **Rotation axes**: Lines about which the square can be rotated by 90°, 180°, and 270° without changing its appearance.
• **Reflection planes**: Lines across which the square can be reflected, such as the vertical, horizontal, and diagonal lines.

The eight symmetry operations include the identity operation (E), which leaves the square unchanged. There are three rotations: 90° (R90), 180° (R180), and 270° (R270), and four reflections: vertical axis (σ_v), horizontal axis (σ_h), and the two diagonals (σ_d1, σ_d2).

These symmetry elements and their respective operations are critical in understanding the group's structure.

Subgroups

Subgroups are smaller groups within a larger group that themselves satisfy the group axioms. For the symmetry operations on a square, there are several subgroups:

1. **Identity subgroup**: Contains only the identity operation (E).
2. **Rotation subgroup**: Includes {E, R90, R180, R270}.
3. **Reflection subgroups**: Examples include {E, σ_v, σ_h, σ_d1}, each containing the identity and some reflections.

Each of these subgroups is closed under the group operation and contains the identity element. They help in analyzing the structure and properties of the overall group of symmetry operations.

Cosets

Cosets are a way of partitioning a group into equal-sized subsets. Given a subgroup H within a group G, a coset is formed by multiplying a fixed element g from G by each element of H. There are two types of cosets:

• **Left cosets**: Formed by multiplying g on the left, denoted gH.
• **Right cosets**: Formed by multiplying g on the right, denoted Hg.

For instance, if we have a subgroup {E, R180} from the symmetry operations of a square, a left coset with element R90 would be {R90, R270}.

Finding the cosets helps in understanding the relationship between different parts of the group and the subgroup structure.