Problem 4 - A Book of Abstract Algebra

Question

If a group \(G\) is generated, say, by \(a, b\), and \(c\), then a set of equations involving \(a, b\), and \(c\) is called a set of defining equations for \(G\) if these equations completely determine the table of \(G\). (See end of Chapter 5.) If \(G^{\prime}\) is another group, generated by elements \(a^{\prime}, b^{\prime}\), and \(c^{\prime}\) satisfying the same defining equations as \(a, b\), and \(c\), then \(G^{\prime}\) has the same table as \(G\) (because the tables of \(G\) and \(G^{\prime}\) are completely determined by the defining equations, which are the same for \(G\) as for \(G^{\prime}\) ). Consequently, if we know generators and defining equations for two groups \(G\) and \(G^{\prime}\), and if we are able to match the generators of \(G\) with those of \(G^{\prime}\) so that the defining equations are the same, we may conclude that \(G \cong G^{\prime}\). Prove that the following pairs of groups \(G, G^{\prime}\) are isomorphic. \(G=\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} ; G^{\prime}=\\{e, a, b, c, a b, a c, b c, a b c\\}\) where \(a^{2}=b^{2}=c^{2}=e\) and \((a b)^{2}=(b c)^{2}=(a c)^{2}=e\)

Step-by-Step Solution

Answer

The groups \( G \) and \( G' \) are isomorphic as they share identical defining relations and structures.

Step 1: Understanding the Groups

We have two groups to compare:1. **Group \( G \)**, which is \( \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \).2. **Group \( G' \)**, which consists of elements \( \{e, a, b, c, ab, ac, bc, abc\} \) with the relations: \( a^2 = b^2 = c^2 = e \) and \( (ab)^2 = (bc)^2 = (ac)^2 = e \).Both groups are expected to be isomorphic based on their structures and the defining properties or equations that characterize the group operations.

Step 2: Analyze Defining Equations for \( G \)

The group \( \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \) consists of an external direct product of three cyclic groups of order 2.The defining property is that in each component, the group elements square to the identity (e.g., for \( a \in \mathbb{Z}_{2}, \ a^2 = e \)).Therefore, every element in \( G \) satisfies the relation \( x^2 = e \) for each group component.The entire group is of order \( 2 \times 2 \times 2 = 8 \), containing all possible combinations of group elements from \( \mathbb{Z}_2 \), which is \( \{(e,e,e), (e,e,g), ..., (g,g,g)\} \) where \( g^2 = e \).

Step 3: Define Equations for \( G' \)

In group \( G' \), we are given a presentation with these equations: \( a^2 = b^2 = c^2 = e \). This states that each generator is of order 2.Additionally, we have the relations \( (ab)^2 = (bc)^2 = (ac)^2 = e \).This implies that the elements formed by combining pairs of generators also have order 2.Thus, every element and product of \( G' \) resolves back to \( e \) when squared, ensuring consistency with an Abelian group of order 8.

Step 4: Compare Generators and Relations

Given the definitions: All combinations of the generators in \( G' \) lead to a minimal presentation matching the element count and structure of \( G \).Both groups possess generators satisfying common relations \( a^2 = b^2 = c^2 = e \) within a finite set of order 8.To confirm isomorphism, permutations of generators indicate that the combined group operations in \( G' \) meet the same criteria as those generated in \( G \). Each distinct group element reflects the combination possibilities in \( \mathbb{Z}_{2}^3 \).

Step 5: Conclude Isomorphic Relationship

Since \( G \) and \( G' \) share identical defining equations (every element self-inverts, pairs invert; yielding identical structural properties and an identical number of elements) and all elements arranged similarly (as manipulation reflects identical subgroup structure), \( G \) is isomorphic to \( G' \).Their tables matching confirms \( G \cong G' \).

Key Concepts

Defining Equations in Group Theory

In group theory, defining equations play a crucial role in understanding the structure of a group. A set of defining equations involves a collection of relations or operations among the elements or generators of a group that uniquely determine that group's properties. These equations provide a comprehensive blueprint of how the group behaves and interacts.

When two groups share the same set of defining equations, they are fundamentally similar in how their elements relate to each other. The defining equations set key properties like how the generators of the group operate under the group multiplication.

This makes it easier to study complex groups by understanding underlying patterns between their elements. If two groups can be described by the same defining equations, we can often show they are isomorphic, meaning they are structurally the same at a fundamental level.

Understanding Cyclic Groups

A cyclic group is a group that can be generated by a single element. This element, called a generator, can create every other element of the group when combined with itself under the group operation.

Consider the example of the group \( \mathbb{Z}_3 \), which is generated by an element like 1. When we apply the group operation of addition under \( \mod 3 \), this single element 1 can generate all of the group's elements: 0, 1, and 2.
That means for any positive integer \( n \), the cyclic group of order \( n \), denoted \( \mathbb{Z}_n \), consists of elements \( \{0, 1, ..., n-1\} \) under addition modulo \( n \).

Cyclic groups are simple yet extraordinarily useful in understanding more complex group structures. Despite their simplicity, they form the building blocks for many other types of groups when considering products or combinations.

Exploring the Nature of Abelian Groups

An Abelian group is a type of group where the operation is commutative, meaning the order in which you combine elements doesn't matter. If you swap any two elements, the result remains the same: \( a \cdot b = b \cdot a \).

This property leads to simplicity and flexibility when handling group operations. For example, the integers under addition form an Abelian group because \( a + b = b + a \) for any integers \( a \) and \( b \).

In our specific context, both groups have elements that square back to the identity element \( e \), and every combination of these elements respects the commutative law.

• This is shown in the exercise with the group \( G' \) by presenting relations: \( a^2 = b^2 = c^2 = e \) and \( (ab)^2 = (bc)^2 = (ac)^2 = e \), indicating that any interaction of the group elements returns you to the starting point, the identity.

Recognizing a group as Abelian helps in predicting behavior and reduces the potential complexity when examining group equations.

Role of Group Generators

Group generators are cornerstones in the formation of a group, as they are the elements from which every other element of the group can be constructed. For a given group, a set of generators is noted when you can express every element in the group as a combination of these generators.

To illustrate, in the group \( G' \) from the exercise, generators include \( a, b, \) and \( c \). These elements fulfill the role of building blocks as combinations of them through the group operation result in the entire group:

• The set \( \{e, a, b, c, ab, ac, bc, abc\} \) displays all possible products, demonstrating how the generators suffuse the structure of \( G' \).

The concept of a generator is fundamental in discerning the nature and capabilities of a group, as they unveil the structure and facilitate comparisons with other groups through common defining equations.