Problem 33 - Algebra: Pure and Applied

Question

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=S_{3} \quad S=\\{(12),(23)\\} $$

Step-by-Step Solution

Answer

Construct a digraph with nodes from \(S_3\) and edges from \( (12) \) and \( (23) \). Define relations: \((12)^2 = e\), \((23)^2 = e\), \((12)(23)(12) = (123)\), and \((23)(12)(23) = (132)\).

Step 1: Identify the Group Elements of \(S_3\)

The symmetric group \(S_3\) consists of all permutations of three elements. The elements are: the identity \(e\), \((12)\), \((13)\), \((23)\), \((123)\), and \((132)\).

Step 2: Define the Generators \(S\) and Apply Them to \(S_3\)

The generators are \((12)\) and \((23)\). These generators allow us to reach each element of \(S_3\) starting from the identity by applying these permutations iteratively.

Step 3: Construct the Cayley Digraph for \(S_3\) with Generating Set \(S\)

Create a directed graph (digraph) where each vertex represents an element of \(S_3\). Draw directed edges based on the operation of the generators:- From any node \(g\), draw an edge to \(g \cdot (12)\) and \(g \cdot (23)\).- For example, from the identity \(e = ()\), we draw an edge to \((12)\) and \((23)\).- Continue this process for all nodes ensuring all group elements are connected.

Step 4: Specify the Defining Relations of \(S_3\)

The defining relations for \(S_3\) with the given generators \((12), (23)\) are:1. \((12)^2 = e\)2. \((23)^2 = e\)3. \((12)(23)(12) = (123)\)4. \((23)(12)(23) = (132)\)

Key Concepts

Symmetric Group S3

The symmetric group \(S_3\) is an important concept in group theory. It consists of all possible permutations of three distinct objects.
This group has six elements, which include:

• The identity permutation \(e\) which does nothing to the set.
• The transpositions \((12), (13), (23)\) which swap two elements.
• The cyclic permutations \((123)\) and \((132)\) which rearrange the entire set in cycles.

Each of these permutations represents a way of arranging or rearranging three items, demonstrating the diversity of operations within \(S_3\). Understanding these elements helps lay the foundation for constructing Cayley digraphs and exploring their algebraic properties.

Group Generators

In group theory, the concept of generators is crucial for understanding how groups function. A set of generators can help describe the entire group by using these generators in various combinations.
In the context of \(S_3\), the generators \((12)\) and \((23)\) are used. With these generators:

• Every element of \(S_3\) can be reached from the identity element \(e\) by a sequence of these operations.
• \((12)\) swaps the first two elements.
• \((23)\) swaps the last two elements.

This means that by iteratively applying these generators, we can express any permutation of the group. This ability to generate the whole group makes these elements particularly important.

Defining Relations

Defining relations are equations that relate the generators of a group with each other and with the identity element. They are critical in fully describing the structure of a group.
For \(S_3\) with generators \((12)\) and \((23)\), the defining relations include:

• \((12)^2 = e\) and \((23)^2 = e\) because each generator is its own inverse.
• \((12)(23)(12) = (123)\) which shows how combinations of generators produce other elements of the group.
• \((23)(12)(23) = (132)\) that also helps in describing the cycling behavior of permutations.

These relations provide constraints and rules governing the group structure, ensuring it is well-defined and consistent.

Permutation Group

A permutation group is a set of permutations that can be performed on a set and forms a group under the operation of composition.
The symmetric group \(S_3\) is a classic example of a permutation group because it includes all permutations for three elements. Key characteristics include:

• Closure: If you compose any two elements of the group, the result is still an element of the group.
• Associativity: The order of applying permutations doesn't change the final outcome, as long as the sequence is preserved.
• Identity: There exists a neutral element \(e\) that doesn't change any element it is applied to.
• Inverses: Each permutation has an inverse permutation that "undoes" it, resulting in the identity.

Understanding permutation groups provides insights into symmetry and order, which are fundamental concepts in abstract algebra.

Directed Graphs

Directed graphs, or digraphs, are graphs where the edges have a direction associated with them. In the context of Cayley digraphs, they represent group elements and the actions of generators.
For \(S_3\) with the generating set \(\{(12), (23)\}\):

• Vertices represent elements of the group.
• Directed edges represent the application of generators.
• An edge from node \(g\) to \(g \cdot (12)\) or \(g \cdot (23)\) showcases how you can "move" from one group element to another using a generator.

This graphical representation helps visualize the structure and dynamics of the group, making it easier to understand the interactions between different elements.