Question

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=\mathbb{Z}_{6}, \quad S=\\{1\\} $$

Short answer

Cayley digraph: a cycle of length 6 with directed edges labeled by 1; relation is \( 1^6 = 0 \).

Step-by-step solution

Identify the Group Structure

The group \( G = \mathbb{Z}_6 \) consists of elements \( \{ 0, 1, 2, 3, 4, 5 \} \) with addition modulo 6 as the operation. This is a cyclic group of order 6.

Describe the Generating Set

The generating set \( S = \{ 1 \} \) means that we can generate all elements of \( G \) by repeatedly adding 1 (mod 6) to any element of \( G \).

Construct the Cayley Digraph

Since our generator is \( 1 \), draw vertices for each element of \( G \): 0, 1, 2, 3, 4, and 5. For each element \( n \), draw a directed edge from \( n \) to \( n+1 \mod 6 \). You will have a directed cycle covering all of the elements of the group.

Define Relations and Verify

Since \( 1 \) generates all elements through repeated addition, we have the relation \( 1^6 = 0 \) in \( \mathbb{Z}_6 \). No additional relations are needed as \( 1 \) naturally traverses the entire group in a cycle.

Key concepts

Understanding Cyclic Groups

Cyclic groups are fascinating entities in group theory. They are characterized by the property that a single element, known as the generator, can produce all elements of the group through repeated operations. These operations depend on the group structure, such as addition or multiplication. For instance, consider the cyclic group \( \mathbb{Z}_6 \).

• This group consists of six elements \( \{ 0, 1, 2, 3, 4, 5 \} \).
• The operation used is addition modulo 6.

Using the element 1 as a generator, you can derive every other element in the group by successively adding 1 and taking results modulo 6, resulting in the structure of a cycle.

Defining a Generating Set

A generating set is a crucial component in understanding how a group is formed. For a group \( G \), a generating set \( S \) consists of one or more elements that can be combined to form every element of \( G \).

In our example with \( \mathbb{Z}_6 \), the generating set \( S = \{1\} \) indicates that the element 1 is powerful enough on its own to generate the whole group. Here’s how it works:

• Add 1 to 0 to get 1.
• Add 1 again to get 2, then continue until you cycle back to 0 modulo 6.

A generating set simplifies understanding the structure and capability of the group by emphasizing key elements that can construct the entire system.

The Role of Modular Arithmetic

Modular arithmetic plays a crucial role in group theory, especially in constructing cyclic groups like \( \mathbb{Z}_6 \). It involves calculations that "wrap around" after a particular value, known as the modulus, is reached.

In the group \( \mathbb{Z}_6 \):

• The elements 0 through 5 reoccur in a loop when you apply modular arithmetic with 6.
• Beginning at any point and continually adding 1 will move you through all elements, demonstrating the 'wrap around' nature of modular arithmetic.

This concept not only defines how group elements interact but also ensures that no matter how large the results, they remain bounded within the group elements, preserving group structure.

Basics of Group Theory

Group theory is a branch of mathematics dedicated to studying algebraic structures known as groups. A group is defined by a set equipped with an operation that combines two elements to form another element within the same set, following four key properties:

• Closure: Any two elements in the group can be combined under the group operation to form another element of the same group.
• Associativity: The group operation is associative, meaning that the grouping of operations does not affect the result.
• Identity Element: There exists an element in the group which, when combined with any element of the group, leaves it unchanged.
• Inverse Element: Every element in the group has an inverse within the group that combines with it to yield the identity element.

By understanding these properties, one can explore diverse mathematical constructions, including cyclic groups, which inherently exhibit these fundamental principles.