Subgroup and Order of a Group are fundamental concepts in group theory. A subgroup is a subset of a group that itself satisfies all the properties of a group under the same operation. The order of a group is the total number of elements present in the group.
Example: The set {0, 2, 4} is a subgroup of {0, 1, 2, 3, 4, 5} under addition modulo 6. The order of the subgroup is 3, while the order of the original group is 6.
Question 1: Find the order of the group G = {0, 1, 2, 3, 4, 5} under addition modulo 6.
The order of a group is the total number of elements in the group.
G = {0, 1, 2, 3, 4, 5} contains six elements.
Therefore, ∣G∣ = 6
Question 2: Find the order of the element 3 in the group (Z9, +)
Solution:
The order of an element is the smallest positive integer n such that n⋅3 ≡ 0
Checking successive multiples: 3 = 3 , 2×3 = 6 , 3 × 3 = 9 = 0(mod 9)
The smallest positive integer satisfying the condition is 3.
Therefore, ord(3) = 3
Question 3: Consider the subgroup H = {0, 3, 6} of the group (Z9, +). Find the order of H.
Solution:
The order of a subgroup equals the number of elements it contains.
The subgroup H = {0,3,6} contains three elements.
Hence, ∣H∣ = 3
Question 4: A finite group G has order 20. Can it have a subgroup of order 8?
Solution:
By Lagrange's Theorem, the order of every subgroup must divide the order of the group.
Here, ∣G∣ = 20
Check whether 8 divides 20: 20 ÷ 8 = 2.5
Since 8 is not a divisor of 20, such a subgroup cannot exist.
Practice Problems
Problem 1: Find the order of the group G = {0, 1, 2, 3, 4, 5, 6, 7} under addition modulo 8.
Problem 2: Find the order of the element 4 in the group (Z12,+)
Problem 3: The subgroup H = {0, 4, 8} belongs to the group (Z₁₂, +). Determine the order of H.
Problem 4: A finite group has order 30. Determine whether a subgroup of order 12 can exist. Justify your answer using Lagrange's Theorem.
