Problem 4 - 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. \(P_{2}\) and \(P_{3} / K\), where \(K=\\{\phi,\\{3\\}\\} .\) [HINT: Consider the function \(f(C)=\) \(C \cap\\{1,2\\} . P_{3}\) is the group of subsets of \(\\{1,2,3\\}\), and \(P_{2}\) of \(\left.\\{1,2\\} .\right]\)

Step-by-Step Solution

Answer

P_2 and P_3/K are isomorphic, confirmed via the fundamental homomorphism theorem.

Step 1: Understand the groups involved

We have two groups: \(P_2\), which is the group of subsets of \(\{1, 2\}\), and \(P_3/K\), where \(P_3\) is the group of subsets of \(\{1, 2, 3\}\). \(K\) is a normal subgroup of \(P_3\), specifically \(K = \{\emptyset, \{3\}\}\). We need to show these two groups are isomorphic.

Step 2: Define the function for homomorphism

Consider the function \(f(C) = C \cap \{1,2\}\) from \(P_3\) to \(P_2\). This function maps a subset of \(\{1,2,3\}\) to a subset of \(\{1,2\}\) by taking the intersection with \(\{1,2\}\). For example, \(f(\{1,3\}) = \{1\}\) and \(f(\{2,3\}) = \{2\}\).

Step 3: Check homomorphism properties

The function \(f\) is a homomorphism because for any two subsets \(A, B \in P_3\), \(f(A \cup B) = (A \cup B) \cap \{1,2\} = (A \cap \{1,2\}) \cup (B \cap \{1,2\}) = f(A) \cup f(B)\). Thus, \(f\) preserves the group operation (union).

Step 4: Determine the kernel of the homomorphism

The kernel \(\ker(f)\) consists of all subsets \(C\) in \(P_3\) for which \(f(C) = C \cap \{1,2\} = \emptyset\). These subsets are either \(\emptyset\) or \(\{3\}\) because any other subset would intersect \(\{1,2\}\) non-trivially. Therefore, \(\ker(f) = \{\emptyset, \{3\}\} = K\).

Step 5: Apply the fundamental homomorphism theorem

The fundamental homomorphism theorem states that if \(f: G \rightarrow G'\) is a group homomorphism with kernel \(K\), then \(G/K \cong f(G)\). Here, \(P_3/K \cong \text{Im}(f)\), and since \(f(P_3) = P_2\), we have \(P_3/K \cong P_2\).

Step 6: Define and verify group tables

Define the group table for \(P_2\): it contains the elements \(\emptyset, \{1\}, \{2\}, \{1,2\}\). For \(P_3/K\), the representatives are \(K\), \(\{1\}K = \{1\}\), \(\{2\}K = \{2\}\), and \(\{1,2\}K = \{1,2\}\). Check that both tables when completed by union operation on these subsets match, confirming the isomorphism.

Key Concepts

Isomorphism

An isomorphism is a special kind of mapping between groups that shows a strong form of equivalence. Two groups, say \( G \) and \( H \), are isomorphic if there exists a bijective (one-to-one and onto) function \( f: G \to H \) that respects the group operations. This means that for any elements \( a, b \) in \( G \), the equation \( f(a \cdot b) = f(a) \cdot f(b) \) holds true. By respecting the group operations, the isomorphism indicates that the two groups have the exact same structure even though their elements or operations may appear different on the surface.

• An isomorphism shows that two mathematical structures are fundamentally the same in terms of group theory.
• It helps in simplifying the study of groups by allowing us to work with a different but equivalent group that is easier to understand.

Understanding the concept of isomorphism is crucial in group theory as it reveals deep connections between seemingly unrelated groups, allowing us to translate knowledge from known group structures to new ones.

Group Theory

Group theory is a fundamental part of modern algebra concerned with studying mathematical groups. A group is defined as a set equipped with a binary operation that satisfies four key properties: closure, associativity, identity, and inversibility. These properties ensure that the group behaves in a predictable and structured way.

• Closure: For any two elements \( a \) and \( b \) in a group \( G \), the result of the operation, \( a \cdot b \), must also be in \( G \).
• Associativity: The group operation must satisfy \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for any elements \( a, b, c \) in \( G \).
• Identity: There exists an identity element \( e \) in \( G \) such that for any element \( a \) in \( G \), the equation \( e \cdot a = a \cdot e = a \) holds true.
• Inverse: For each element \( a \) in \( G \), there exists an inverse element \( a^{-1} \) in \( G \) such that \( a \cdot a^{-1} = a^{-1} \cdot a = e \).

The study of groups is not limited to numbers but also applies to other mathematical objects. Group theory is fundamental in numerous areas of mathematics and has applications in physics, cryptography, and many other fields.

Normal Subgroup

A normal subgroup is a special type of subgroup that plays a central role in the study and application of group theory, particularly in the context of group homomorphisms. A subgroup \( N \) of a group \( G \) is normal if it satisfies the condition \( gNg^{-1} = N \) for every element \( g \) in \( G \). This condition essentially means that the subgroup remains the same no matter how it is conjugated by elements of the larger group.

• Normal subgroups are crucial because they allow the construction of quotient groups, enabling the use of the fundamental homomorphism theorem.
• The notation \( N \trianglelefteq G \) is commonly used to denote that \( N \) is a normal subgroup of \( G \).

In the context of the exercise, \( K = \{ \emptyset, \{3\} \} \) is a normal subgroup of \( P_3 \). This allows us to consider the quotient group \( P_3 / K \), which simplifies many problems in group theory and is instrumental in proving isomorphisms using the fundamental homomorphism theorem.

Group Homomorphism

A group homomorphism is a function between two groups that respects their structure by preserving the group operation. Formally, a function \( f: G \to H \) is a homomorphism if, for every pair of elements \( a, b \in G \), the equation \( f(a \cdot b) = f(a) \cdot f(b) \) holds true. The preservation of the operation ensures that the image of the elements of \( G \) under \( f \) in \( H \) maintains the structural features of \( G \).

• A crucial concept for group homomorphisms is the kernel, defined as \( \ker(f) = \{ g \in G \mid f(g) = e_H \} \), where \( e_H \) is the identity in \( H \).
• The kernel measures the "failure" of injectivity in \( f \). A kernel being just the identity indicates that the homomorphism is injective (one-to-one).
• Surjectivity (onto) ensures that every element of \( H \) is the image of some element from \( G \), making \( f \) a surjective homomorphism when its image is equal to \( H \).

In this exercise, the function \( f: P_3 \rightarrow P_2 \) defined by intersection with \( \{1,2\} \) demonstrates these properties, with \( \ker(f) = \{ \emptyset, \{3\} \} = K \), allowing the application of the fundamental homomorphism theorem.