A Cyclic Group is a group in which every element can be generated by repeatedly applying the group operation to a single element, called the generator.
The group {0, 1, 2, 3, 4, 5} under addition modulo 6 is a cyclic group because every element can be generated by repeatedly adding 1 (or 5) modulo 6.
Question 1: Construct the Cayley Table for (Z4,+) and Determine Whether It Is a Cyclic Group
The elements of (Z4,+)are Z4 = {0,1,2,3}
The operation is addition modulo 4.
Step 1: Construct the Cayley Table
+₄ 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 Step 2: Identify the Identity Element
The row and column corresponding to 0 reproduce all group elements.
Therefore, Identity Element=0
Step 3: Determine Whether the Group is Cyclic
Using the element 1, 1, 1+1 = 2, 2+1 = 3, 3+1 = 0
All elements are generated.
Similarly, 3, 3+3 = 2, 2+3 = 1, 1+3 = 0 also generates every element.
Hence, the generators are 1, 3
Therefore, (Z4,+) is a cyclic group.
Question 2: Construct the Cayley Table for (Z5,+). Verify Whether the Table is Symmetric and Find All Generators.
Solution
The elements are Z5 = {0,1,2,3,4} using addition modulo 5.
Step 1: Construct the Cayley Table
+₅ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 Step 2: Check Symmetry: Since, a+b ≡ b+a(mod5)
the entries are mirrored across the main diagonal.
Therefore, The Cayley table is symmetric
This shows that the group is abelian.
Step 3: Find the Generators
Starting from each element:
- 1 → 2 → 3 → 4 → 0
- 2 → 4 → 1 → 3 → 0
- 3 → 1 → 4 → 2 → 0
- 4 → 3 → 2 → 1 → 0
Each generates every element.
Hence, Generators = {1, 2, 3, 4}
Thus, (Z5,+) is a cyclic group.
Practice Questions
1. Determine whether the group (Z8,+)is cyclic. If it is cyclic, find all of its generators.
2. Find the order of each element in the cyclic group (Z10,+). Also, identify all the generators of the group.
3. Construct the Cayley table for the group (Z3,+) under addition modulo 3. Identify the identity element and verify whether the table is symmetric.
4.Construct the Cayley table for the group (Z5,+) under addition modulo 5. Using the table, determine whether the group is cyclic, identify all its generators, and state whether the group is abelian.
