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Magazine Chapter 7: Problem 16
Draw a Cayley digraph for \(Z_{4}\) taking as generating set \(S=\\{2,5]\).
Short Answer
Expert verified
The Cayley digraph of \(Z_4\) with \(S=\{2, 5\}\) is a graph with nodes 0, 1, 2, 3 and directed edges formed by adding 1 and 2 modulo 4.
Step by step solution
01
Understanding Cayley Digraph
A Cayley digraph of a group is a directed graph representing the elements of the group and the connections between them based on a specific generating set. For the group \(Z_4\), we consider the elements \{0, 1, 2, 3\}, representing integers modulo 4. The generating set \(S = \{2, 5\}\) tells us which operations define the edges in the digraph.
02
Identify Generators and Reduced Elements
The elements of the generating set \(S = \{2, 5\}\) must be considered modulo 4. Since 5 modulo 4 is 1 (because 5 div 4 leaves a remainder of 1), the effective generating set becomes \(\{2, 1\}\). This simplifies our task to draw arrows based on these steps in the group \(Z_4\).
03
Draw Nodes of the Cayley Digraph
For the group \(Z_4\), we draw four nodes labeled \(0, 1, 2, 3\). These will represent the elements of \(Z_4\).
04
Add Arrows for Generator 2
Draw directed arrows from each node to the node that results from adding 2 modulo 4:
- From node 0, add 2 to reach node 2.
- From node 1, add 2 to reach node 3.
- From node 2, add 2 to reach node 0 (4 modulo 4 is 0).
- From node 3, add 2 to reach node 1.
05
Add Arrows for Generator 1
Draw directed arrows from each node to the node that results from adding 1 modulo 4:
- From node 0, add 1 to reach node 1.
- From node 1, add 1 to reach node 2.
- From node 2, add 1 to reach node 3.
- From node 3, add 1 to reach node 0 (4 modulo 4 is 0).
06
Complete the Digraph
With all arrows for both generators added, complete the diagram with all connections:
- Connect arrows between nodes as follows:
- Node 0 to nodes 1 (for generator 1) and 2 (for generator 2).
- Node 1 to nodes 2 (for generator 1) and 3 (for generator 2).
- Node 2 to nodes 3 (for generator 1) and 0 (for generator 2).
- Node 3 to nodes 0 (for generator 1) and 1 (for generator 2).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Group Theory
Group theory is a mathematical framework for studying structures known as groups, which consist of a set of elements and an operation that combines any two elements to form a third element within the set. This operation must satisfy four fundamental properties known as the group axioms:
- Closure: For any two elements in the group, their combination results in another element within the group.
- Associativity: The order in which operations are performed does not affect the outcome, i.e., if \(a\), \(b\), and \(c\) are elements of the group, then \((a * b) * c = a * (b * c)\).
- Identity Element: There exists an element in the group, known as the identity, which when combined with any element of the group returns the same element.
- Inverse Element: Every element in the group has an inverse such that when it is combined with the element, it results in the identity element.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, known as the modulus. It's sometimes described as "clock arithmetic" because it behaves like the hours on a clock.
- Given a modulus \(n\), the expression \(a \mod n\) means the remainder of the division of \(a\) by \(n\).
- In the group \(Z_4\), calculations are done with modulus 4, so every result is reduced to one of the values \{0, 1, 2, 3\}.
- This reduction is crucial in simplifying calculations and has applications in computer science, cryptography, and number theory.
Generating Set
A generating set in the context of group theory is a subset of a group such that every element of the group can be expressed as a combination (using the group operation) of finitely many elements of this subset.
- For example, if \(S = \{2, 5\}\) is a generating set of the group \(Z_4\), then every element of \(Z_4\) can be reached using operations involving just these numbers.
- The generating set simplifies understanding the structure and operations within the group.
- In our exercise, the generating set is reduced based on modular arithmetic to \(\{2, 1\}\), where the number 5 becomes 1 mod 4.
