Problem 3 - 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. \(G=D_{4} ; G^{\prime}=\left\\{e, a, b, a b, a b a,(a b)^{2}, b a, b a b\right\\}\) where \(a^{2}=b^{2}=e\) and \((a b)^{4}=e\).

Step-by-Step Solution

Answer

The groups \(G = D_4\) and \(G'\) are isomorphic because they satisfy the same defining relations.

Step 1: Identify the Defining Equations for G

The dihedral group of order 8, denoted as \(D_4\), is generated by two elements, a rotation \(r\) and a reflection \(s\). The defining relations are \(r^4 = e\), \(s^2 = e\), and \(srs = r^{-1}\). However, you can alternatively express these as \(a^2 = e\), \(b^2 = e\), and \((ab)^4 = e\) by substituting \(a = s\) and \(b = sr\).

Step 2: Analyze the Given Group G'

For the group \(G^\prime\), it consists of the elements \(\{e, a, b, ab, aba, (ab)^2, ba, bab\}\), satisfying \(a^2 = b^2 = e\) and \((ab)^4 = e\). These are the same relations seen in the defining equations we've identified for \(D_4\).

Step 3: Match the Generators of G and G'

Each of the elements in the given set for \(G^\prime\) can be seen to satisfy the same relations as those for \(D_4\). By mapping the elements correspondingly (keeping in mind the relations like \((ab)^{-1} = b^{-1}a^{-1} = ba\)), it ensures the relations \((ab)^4 = e\), \(a^2 = b^2 = e\) are satisfied, just as in \(D_4\).

Step 4: Conclude Isomorphism

Since both \(G\) and \(G^\prime\) satisfy the same defining equations and generators can be matched, they share the same group structure. Thus, we conclude that \(G \cong G^\prime\).

Key Concepts

Group Theory

Group theory is a mathematical field that studies algebraic structures known as groups. A group is a set equipped with an operation that combines any two elements to form a third element, adhering to four core properties:

• Closure: The operation on any two elements of the group yields another element within the group.
• Associativity: The operation is associative, meaning (a * b) * c = a * (b * c).
• Identity Element: There is an element within the group that remains unchanged when combined with any other element of the group.
• Inverse Element: Each element in the group has an inverse; when an element is combined with its inverse, the result is the identity element.

Group theory is foundational in the study of symmetries as it provides a formal way to discuss and analyze the mathematical structure and symmetries of objects.

Defining Equations

Defining equations, often referred to as presentation relations, are a set of equations that uniquely describe a group in terms of its generators.
These equations include all the relations that must be true among the generators to entirely define the group structure. For instance, for a group generated by elements \( a, b \, \text{and} \, c \), the defining equations might express properties like \( a^2 = e \, \text{(identity)}, \) \( b^2 = e \), and relationships such as \( ab = ba^{-1} \, \text{(commutation relation)}.\)
These equations are crucial in determining if two groups are isomorphic, meaning one-to-one correspondence exists between their elements and operations. Essentially, if two groups share the same generators and defining equations, they are considered structurally identical, or isomorphic.

Dihedral Group

The dihedral group, symbolized as \( D_n \), is a classic example of symmetry group in mathematics. Primarily, it describes the symmetries of a regular polygon with \( n \) sides.
This group consists both of rotational and reflectional symmetries.
For instance, the dihedral group \( D_4 \) relates to the symmetries of a square. It includes:

• 4 rotations (including the identity rotation),
• and 4 reflections.

The group is defined by specific relations among its generators, typically a rotation \( r \) and a reflection \( s \). Standard relations within this group include: \[ r^n = e, \, s^2 = e, \, \text{and} \, srs = r^{-1}.\] These dictate that a full rotation returns to the starting point, and reflecting twice acts as the identity transformation. The structure of these relations helps identify different transformations within the group.

Group Generators

In group theory, generators are elements from which every element of the group can be derived through the group operation. They are fundamental in constructing the entire group by applying the group operation repetitively.
For example, if a group \( G \) is generated by two elements, \( a \, \text{and} \, b, \) then every element in \( G \) can be expressed as a combination of \( a \, \text{and} \, b \).
The significance of group generators extends to identifying groups and their properties. For the dihedral group \( D_4 \), generators can be a rotation \( r \) and a reflection \( s \), allowing one to construct all possible symmetries of a square by combining \( r \) and \( s \) in various sequences and quantities. Understanding these generators and their defined relationships helps in determining how elements interact within the group, aiding in tasks such as solving isomorphisms between different groups.