Problem 1 - 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. \(G=\) the subgroup of \(S_{4}\) generated by \((24)\) and \((1234) ; G^{\prime}=\left\\{e, a, b, b^{2}, b^{3}, a b,\right.\), \(\left.a b^{2}, a b^{3}\right\\}\) where \(a^{2}=e, b^{4}=e\), and \(b a=a b^{3}\).

Step-by-Step Solution

Answer

Groups \(G\) and \(G'\) are isomorphic because they share identical generators and defining relations.

Step 1: Identify Generators and Relations for G

The group \( G \) is a subgroup of \( S_4 \), the symmetric group on 4 elements, generated by the cycles \( (24) \) and \( (1234) \). We need to establish the relations: \( (24)(24) = e \), \( (1234)(1234)(1234)(1234) = e \), and understand how these generators interact.

Step 2: Determine Relations in G based on Elements

In \( S_4 \), the permutation \((24)\) is an involution, meaning it has order 2, so \((24)^2 = e\). The permutation \((1234)\) has order 4 because it cycles four numbers, hence \((1234)^4 = e\). Therefore, the defining relations for \(G\) are \(a^2 = e\), \(b^4 = e\), and \(ab = ba^3\).

Step 3: Identify Generators and Relations for G'

The group \( G' \) is given with a set of elements \{e, a, b, b^2, b^3, ab, ab^2, ab^3\} and relations \(a^2 = e\), \(b^4 = e\), and \(ba = ab^3\). We need to confirm these correspond to the relations in \(G\).

Step 4: Compare Relations in G and G'

Notice that both \(G\) and \(G'\) have the same relations: generators satisfy \(a^2 = e\), \(b^4 = e\), with relations involving \(a\) and \(b\) showing that \(b\) acts like a 4-cycle and \(a\) like a 2-cycle, specifically \(ab = ba^3\).

Step 5: Conclusion about Isomorphism

Since both groups \(G\) and \(G'\) have equivalent generating sets and defining relationships, they have the same group structure. Therefore, \(G\) and \(G'\) are isomorphic, which means \(G \cong G'\).

Key Concepts

Defining Equations

In group theory, defining equations are fundamental to understanding the structure of a group. When a group is generated by specific elements, the defining equations describe how these elements interact with each other. In essence, they set the rules of engagement for the group's operations. These equations are crucial because they completely determine the group's operation table. By specifying these equations, you can predict all possible combinations and products of the group's elements.
For instance, consider a group generated by elements \(a\), \(b\), and \(c\). Defining equations like \(a^2 = e\) and \(b^4 = e\) outline that \(a\) and \(b\) have orders 2 and 4, respectively, meaning you'll return to the group identity element \(e\) after that many applications of the operation. Such equations are what's known as relations. If two groups have the same defining equations, they can be shown to have similar mathematical structures, leading us into the concept of isomorphic groups.

Generators

Generators are elements from which you can build the entire group. Think of them as the building blocks of the group. For any given group, generators allow you to obtain every other element of the group by applying the group's operation repeatedly to these generators.
In the exercise, group \(G\) is generated by the permutations \((24)\) and \((1234)\). This means applying and combining these permutations in various ways can produce any element within the subgroup of the symmetric group \(S_4\) that \(G\) forms. Similarly, group \(G'\) is generated by \(a\) and \(b\), and manipulating these elements according to defined rules reveals the structure of \(G'\).
This shows the power of generators—they simplify the representation of a complex group by reducing it to a few fundamental components.

Isomorphic Groups

Isomorphic groups are groups that, although possibly different in appearance or construction, share the same internal structure. In mathematics, two groups \(G\) and \(G'\) are said to be isomorphic if there is a one-to-one correspondence between their elements that preserves the group operation.
To determine if groups are isomorphic, we typically compare their generating elements and defining equations. When the exercise shows that both \(G\) and \(G'\) have generators that satisfy the same defining equations, it directly implies that the groups have the same structure. The main takeaway is that because these equations govern the operation within the groups, having identical relations means the groups can be transformed into one another by relabeling their elements keeping the operations intact.
This tells us that isomorphism isn’t concerned with the "nature" of the elements, just how they interact mathematically.

Symmetric Group

The symmetric group, often denoted as \(S_n\), is the group of all permutations of \(n\) objects. It plays a pivotal role in group theory due to its comprehensive structure and versatility. Every element of a symmetric group is a permutation, which is essentially a rearrangement of the group's elements.
In our example, \(S_4\) represents the symmetric group on 4 elements. The subgroup \(G\) within \(S_4\) is defined by permutations like \((24)\) and \((1234)\). These cycles suggest the movement of elements under permutations: \((24)\) swaps elements 2 and 4, while \((1234)\) rotates four elements in sequence.
The importance of the symmetric group lies in its application to isomorphisms and its ability to provide a representation of finite groups, making it a cornerstone of abstract algebra.

Permutations in Groups

Permutations describe the rearrangements of elements within a set based on specific patterns. In group theory, a permutation is not just an operation but a transformation that reshuffles elements under a set of rules, maintaining a one-to-one correspondence.
Understanding permutations involves recognizing cycles, such as \((1234)\), which organize elements into layers of interaction—the cycle specifies the order in which elements are mapped to each other. A permutation's properties, such as its order, dictate how often you must apply it to return to the starting position.
In groups, permutations play a pivotal role as they embody the symmetries and underlying structure of the group itself. For instance, in the symmetric group \(S_4\), permutations like \((24)\) exhibit their distinct influence, and as generators, they help define the group structure and relations leading to defining equations and, ultimately, the isomorphism with other groups.