Problem 2 - A Book of Abstract Algebra

Question

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{3} \text { and } \mathbb{Z}_{6} /\langle 3\rangle $$

Step-by-Step Solution

Answer

\( \mathbb{Z}_3 \) and \( \mathbb{Z}_6 / \langle 3 \rangle \) are isomorphic.

Step 1: Understanding the Groups

We need to understand what each group represents. The group \( \mathbb{Z}_3 \) is the set \( \{ 0, 1, 2 \} \) with addition modulo 3. The group \( \mathbb{Z}_6 / \langle 3 \rangle \) is a quotient group formed from \( \mathbb{Z}_6 \) by the subgroup generated by 3, \( \langle 3 \rangle \).

Step 2: Identify the Subgroup

The subgroup \( \langle 3 \rangle \) in \( \mathbb{Z}_6 \) consists of all multiples of 3, which are \( \{0, 3\} \). Hence, \( \mathbb{Z}_6 / \langle 3 \rangle \) has cosets \( \{0, 3\} \) and all integers not a multiple of 3 in \( \mathbb{Z}_6 \): \( \{1\}, \{2\}, \{4\}, \{5\} \) relate to \( \{1, 4\} \) and \( \{2, 5\} \) respectively, giving cosets \( \{0, 3\}, \{1, 4\}, \{2, 5\} \).

Step 3: Cosets of the Quotient Group

The cosets \( \{0, 3\}, \{1, 4\}, \{2, 5\} \) can be relabeled as \( \{0\}, \{1\}, \{2\} \) since only the equivalence class matters. This makes \( \mathbb{Z}_6 / \langle 3 \rangle \) have elements behaving as in \( \mathbb{Z}_3 \).

Step 4: Construct the Isomorphism

Define a homomorphism \( f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3 \) by \( f(x) = x \mod 3 \). The kernel of this map is \( \langle 3 \rangle \), and its image is all of \( \mathbb{Z}_3 \). Hence, by the fundamental homomorphism theorem, \( \mathbb{Z}_6 / \langle 3 \rangle \cong \mathbb{Z}_3 \).

Step 5: Display the Group Tables

The group addition table for \( \mathbb{Z}_3 \) is as follows:\[\begin{array}{c|ccc}+ & 0 & 1 & 2 \\hline0 & 0 & 1 & 2 \1 & 1 & 2 & 0 \2 & 2 & 0 & 1 \\end{array}\]The group operation table for \( \mathbb{Z}_6 / \langle 3 \rangle \) is the same:\[\begin{array}{c|ccc}+ & \{0\} & \{1\} & \{2\} \\hline\{0\} & \{0\} & \{1\} & \{2\} \\{1\} & \{1\} & \{2\} & \{0\} \\{2\} & \{2\} & \{0\} & \{1\} \\end{array}\]

Key Concepts

Group Isomorphism

When studying group theory, you may often compare two groups to determine if they are essentially the same in structure. This concept is known as a **group isomorphism**. An **isomorphism** between two groups, say \( G \) and \( H \), is a bijective function \( \phi: G \rightarrow H \) that preserves the group operation. In simpler terms, this means that \( \phi(g_1 g_2) = \phi(g_1) \phi(g_2) \) for all \( g_1, g_2 \) in \( G \).

• An isomorphism shows that two groups have the same structure.
• Every element in one group corresponds uniquely to an element in the other.

Both groups can be considered essentially identical in terms of their group-theoretic properties, even if their elements or operation modes look different on the surface. Essentially, if two groups are isomorphic, they behave the same, and their group tables will be identical, even if they are formed differently. In the context of the exercise, this is the reason why \( \mathbb{Z}_3 \) and \( \mathbb{Z}_6 / \langle 3 \rangle \) are isomorphic—they have the same group operation table.

Quotient Group

A **quotient group** is a way to construct a new group from an existing group by "collapsing" some of its elements together. If you have a group \( G \) and a normal subgroup \( N \) of \( G \), then the quotient group \( G/N \) is the set of cosets of \( N \) in \( G \). Each coset is a kind of "group chunk," representing all elements of \( G \) that are similar under multiplication by elements of \( N \).
For example, in the context of our original problem, \( \mathbb{Z}_6 / \langle 3 \rangle \) takes the full group of integers modulo 6 and identifies elements that differ by multiples of 3. This collapses the group down into three equivalence classes, which form the elements of the quotient group. The operation of addition on these cosets reflects the operation in the original group, just applied to entire portions of it.

• Quotient groups help in simplifying and understanding the structure of groups.
• They are crucial in the study of group homomorphisms and isomorphisms.

Cosets

**Cosets** are a fundamental concept within group theory that helps in understanding the structure of groups and quotient groups. Given a group \( G \) and a subgroup \( H \) of \( G \), a coset is formed by taking an element \( g \) from \( G \) and combining it with all elements of \( H \).
There are two types of cosets:

• **Left Coset**: Defined as \( gH = \{ gh : h \in H \} \).
• **Right Coset**: Defined as \( Hg = \{ hg : h \in H \} \).

Typically, in the sense of quotient groups, we deal with left cosets. They partition the group into equal-sized, disjoint subsets. The left cosets of \( \langle 3 \rangle \) in \( \mathbb{Z}_6 \) are \( \{ 0, 3 \}, \{ 1, 4 \}, \{ 2, 5 \} \). These cosets correspond to the elements of the quotient group \( \mathbb{Z}_6 / \langle 3 \rangle \). This notion of cosets helps in defining the elements of a quotient group by capturing the idea that different elements can represent equivalent groups of transformations.

Group Theory

**Group theory** is a branch of mathematics that studies the algebraic structures known as groups. At its heart, it is concerned with understanding symmetry, structure, and operations within a set.
In group theory, a group is defined as a set combined with an operation that satisfies four key properties:

• **Closure**: For every two elements \( a \) and \( b \) in the group, the result of the operation, \( a*b \), is also in the group.
• **Associativity**: For three elements \( a, b, \) and \( c \), the equation \((a*b)*c = a*(b*c)\) holds.
• **Identity Element**: There exists an element in the group that is the identity with respect to the operation, meaning \(a*e = e*a = a\) for any element \( a \).
• **Inverses**: For each element \( a \) in the group, there is an element \( b \) such that \( a*b = b*a = e \), where \( e \) is the identity element.

Group theory is foundational in many areas of mathematics and applies to multiple disciplines, including physics, chemistry, and computer science. Its versatility and power lie in its ability to abstractly capture the essence of symmetry and systematize the way elements relate and combine.