Problem 2 - 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. \(G=S_{3} ; G^{\prime}=\\{e, a, b, a b, a b a, a b a b\\}\) where \(a^{2}=e, b^{2}=e\), and \(b a b=a b a\).

Step-by-Step Solution

Answer

The groups \( G = S_3 \) and \( G' = \{e, a, b, ab, aba, abab\} \) are isomorphic.

Step 1: Define the Groups

The group \( G = S_3 \) is the symmetric group of degree 3, which includes all permutations of three elements. It has 6 elements: \( \{ e, (12), (13), (23), (123), (132) \} \). The group \( G' = \{ e, a, b, ab, aba, abab \} \) is presented with generators \( a \) and \( b \) and relations \( a^2 = e, b^2 = e, ba = ab \). We need to show that these groups are isomorphic.

Step 2: Verify Orders and Presentations

Check the orders of both groups. \( G \) has 6 elements as it is the symmetric group \( S_3 \). Similarly, \( G' \) also has 6 elements as defined. The given generators and relations define a group of size 6: \( a^2 = e, b^2 = e \) (these imply \( a \) and \( b \) are each of order 2), and \( ba = aba \) allows no further reduction in group order. Thus, both groups are of order 6.

Step 3: Check the Group Presentation

The presentation for \( S_3 \) can be defined as: generators \( \sigma = (123) \) and \( \tau = (12) \), with relations \( \sigma^3 = e \), \( \tau^2 = e \) and \( (\tau\sigma)^2 = e \). Compare the relations: let \( a = \tau \), \( b = \tau\sigma \). Then, \( a^2 = e \), \( b^2 = a(\tau\sigma)^2a = e \), \( ba = ab \). These match the relations \( a^2 = e \), \( b^2 = e \), and \( bab = aba \) given for \( G' \).

Step 4: Establish the Isomorphism

The relabeling of elements in \( G' \) to match those in \( S_3 \) is consistent: Assign \( a \to (12) \) and \( b \to (123) \) based on verified presentation comparisons. Then, construct all elements of \( G' \) using these assignments, which will match those in \( S_3 \). Therefore, by establishing a one-to-one function respecting group operations, it's assured that \( G' \) is isomorphic to \( S_3 \). Thus, \( G \cong G' \).

Key Concepts

Symmetric Group

The symmetric group, denoted as \( S_n \), is a fundamental concept in group theory. It consists of all possible permutations of \( n \) objects. When \( n = 3 \), you have \( S_3 \), which includes all ways to rearrange three distinct items. This group has six elements because you can arrange three items in \( 3! = 6 \) different ways.
Elements in \( S_3 \) include:

• the identity permutation \( e \), which leaves everything in place,
• transpositions like \( (12) \), \( (13) \), and \( (23) \), which swap two elements,
• and cycles like \( (123) \) and \( (132) \), which rotate all three elements.

Each permutation has a specific order, determined by how many times it must be applied to return to the starting arrangement. For instance, all transpositions are of order 2, reflecting the number of swaps needed to revert them. Understanding the symmetric group is vital because it serves as a building block for other group concepts.

Group Presentation

A group presentation gives us a formal way to describe a group using a set of generators and specific relations among them. In essence, instead of listing all elements, you can express a group through simplified terms.
The presentation specifies:

• a set of \'generators\', which are elements that can be combined to create every other element in the group,
• and \'relations\', equations that define how these generators interact.

For example, a presentation of \( G' = \{ e, a, b, ab, aba, abab \} \) suggests generators \( a \) and \( b \) with relations such as \( a^2 = e \) and \( b^2 = e \). Instead of listing out all permutations, this presentation succinctly provides an equivalent description of the group.

Group Generators

Generators form the backbone of a group. They are specific elements from which every other element of the group can be derived through combinations. This is particularly useful when dealing with complex groups where listing every element would be cumbersome.
In group \( G' \), the generators \( a \) and \( b \) allow us to create other elements like \( ab \), \( aba \), and so on, by following the rules set by the group. The most notable points about generators are:

• They help define the group's structure using the fewest elements possible,
• Their interactions are governed by the group’s defining relations.

Understanding how to use and apply generators can transform and simplify complex algebraic expressions and calculations within the group.

Defining Equations

Defining equations, also known as relations, are essential as they impose specific rules or conditions on the group generators. These equations are crucial because they ensure that the elements behave in particular ways, outlining the structure of the group.
In our example, for both \( G = S_3 \) and \( G' \), the defining equations are:

• \( a^2 = e \), meaning using the generator \( a \) twice gives you the identity,
• \( b^2 = e \), a similar rule for the generator \( b \),
• \( ba = aba \), which dictates how \( a \) and \( b \) interact.

Such equations ensure the group's overall consistency and uniqueness. They are integral in proving group isomorphism, as identical defining equations in different groups suggest they share the same structure.