Problem 3 - A Book of Abstract Algebra

Question

Describe the subgroup of \(\mathbb{Z}_{12}\) generated by 6 and \(9 .\)

Step-by-Step Solution

Answer

The subgroup generated by 6 and 9 in \(\mathbb{Z}_{12}\) is \(\{0, 3, 6, 9\}\).

Step 1: List Elements of \(\mathbb{Z}_{12}\)

The group \(\mathbb{Z}_{12}\) contains the elements \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \). This group has addition modulo 12 as its operation.

Step 2: Understand Generation Concept

A subgroup generated by elements is the set of all possible sums using those elements and their multiples. In this case, we generate using both 6 and 9 with addition modulo 12.

Step 3: Combine Multiples of 6 and 9

Compute multiples of 6 and 9 modulo 12. These include:Multiples of 6: \( 0, 6, 12(=0) \dots \) modulo 12 yields \( \{0, 6\} \).Multiples of 9: \( 0, 9, 18(=6) \dots \) modulo 12 yields \( \{0, 9, 6\} \).

Step 4: Perform Linear Combinations Modulo 12

Consider all combinations: \(a\cdot6 + b\cdot9\) with \(a, b \in \mathbb{Z}\). Modulo 12,- \(1\times6 + 0\times9 = 6\)- \(0\times6 + 1\times9 = 9\)- \(1\times6 + 1\times9 = 15 \equiv 3 \mod 12 \)- \(2\times6 = 12 \equiv 0 \mod 12 \)- \(2\times6 + 1\times9 = 21 \equiv 9 \mod 12 \)Reaching elements: \( \{0, 3, 6, 9\} \).

Step 5: Identify Subgroup

The set \( \{0, 3, 6, 9\} \) closes under addition modulo 12 and is generated by 6 and 9, meaning if you add any two elements within the set and reduce modulo 12, you stay within the set. Thus, it is the subgroup generated by 6 and 9.

Key Concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers that considers the remainder when a number is divided by another number. It's often used in group theory, cryptography, and computer science.

In modular arithmetic, we say numbers are congruent if they have the same remainder when divided by a given modulus. For example, in \(\mathbb{Z}_{12}\), the numbers 10 and 22 are congruent because both leave a remainder of 10 when divided by 12. We write this as \(10 \equiv 22 \mod 12\).

• The operation used is addition, subtraction, or multiplication, followed by taking the result modulo a number (such as 12).
• It creates a 'circular' number system, where after reaching the modulus, the numbers wrap around back to zero.

In the original problem, modular arithmetic helps us understand how to generate a subgroup within \(\mathbb{Z}_{12}\). It involves computing the multiples of elements like 6 and 9 and then evaluating them under modulo 12, simplifying the numbers into a smaller, confined set.

Subgroups

A subgroup is a subset of a larger group that is, by itself, a group under the same operation. Think of it as a 'smaller' group inside a 'big' group that behaves exactly like the big group regarding its operations.

In mathematical terms, for a non-empty subset \(H\) of a group \(G\) to be a subgroup:

• \(H\) must be closed under the operation defined in \(G\). If \(a, b \in H\), then \(a \cdot b \in H\).
• It must contain the identity element of \(G\).
• Every element in \(H\) must have an inverse that is also in \(H\).

In our example, the subgroup is generated by using linear combinations of the elements 6 and 9 within \(\mathbb{Z}_{12}\). The numbers {0, 3, 6, 9} create a subgroup because they satisfy all the conditions of closure and inverses under modulo 12 addition.

Finite Groups

Finite groups are algebraic structures that consist of a set of elements along with an operation that combines any two elements to form a third element, all while containing a finite number of elements. Group theory studies these structures to understand symmetry and transformations.

• Finite groups must satisfy four conditions: closure, associativity, identity, and invertibility.
• The order of a group is the number of its elements.

For instance, \(\mathbb{Z}_{12}\) is a finite group. It has 12 elements and uses addition modulo 12 as its operation. The subgroup \(\{0, 3, 6, 9\}\) is a finite group because it contains a limited number of elements (4 in total) and satisfies the group properties within the larger group of modulo 12.