Problem 4 - A Book of Abstract Algebra

Question

Let \(G, H\), and \(K\) be groups. Prove the following: If \(f: G \rightarrow H\) is a homomorphism and \(J\) is any subgroup of \(H\), then- $$ f^{-1}(J)=\\{x \in G: f(x) \in J\\} $$ is a subgroup of \(G\). Furthermore, ker \(f \subseteq f^{-1}(J)\).

Step-by-Step Solution

Answer

\(f^{-1}(J)\) is a subgroup of \(G\), and \(\ker f \subseteq f^{-1}(J)\).

Step 1: Confirm Definition of Preimage Set

We begin by understanding that \(f^{-1}(J) = \{x \in G : f(x) \in J\}\) is defined as the set of all elements \(x\) in \(G\) such that when the homomorphism \(f\) is applied to \(x\), the result lies in the subgroup \(J\) of \(H\).

Step 2: Show Non-emptiness

Since \(e_H\), the identity element of \(H\), belongs to every subgroup \(J\), and since \(f(e_G) = e_H\) (because \(f\) is a homomorphism), it follows that \(e_G \in f^{-1}(J)\). Thus, \(f^{-1}(J)\) is non-empty.

Step 3: Confirm Closure Under Group Operation

Consider any \(a, b \in f^{-1}(J)\). Then \(f(a), f(b) \in J\). Since \(J\) is a subgroup, \(f(a)f(b) \in J\). By the homomorphism property, \(f(ab) = f(a)f(b)\). Thus, \(ab \in f^{-1}(J)\), meaning \(f^{-1}(J)\) is closed under the group operation.

Step 4: Confirm Closure Under Inverses

For \(a \in f^{-1}(J)\), \(f(a) \in J\) and since \(J\) is a subgroup, \(f(a)^{-1} \in J\). Given \(f(a^{-1}) = f(a)^{-1}\) by the homomorphism property, it follows that \(a^{-1} \in f^{-1}(J)\). Hence, \(f^{-1}(J)\) is closed under taking inverses.

Step 5: Conclude \(f^{-1}(J)\) is a Subgroup

Having shown that \(f^{-1}(J)\) is non-empty, closed under the group operation, and closed under inverses, \(f^{-1}(J)\) satisfies the subgroup criteria and is therefore a subgroup of \(G\).

Step 6: Show Kernel Inclusion

The kernel of \(f\), denoted \(\ker f\), is the set \(\{x \in G : f(x) = e_H\}\). Since \(e_H \in J\), all elements of the kernel are in \(f^{-1}(J)\). Thus, \(\ker f \subseteq f^{-1}(J)\).

Key Concepts

Preimage of a Subgroup

In group theory, understanding the concept of the preimage of a subgroup is essential when working with homomorphisms. When you have a group homomorphism \( f: G \rightarrow H \), and a subgroup \( J \) of \( H \), the preimage \( f^{-1}(J) \) is the set of elements from group \( G \) that map to elements of \( J \) under \( f \). In mathematical terms, this is expressed as \( f^{-1}(J) = \{x \in G: f(x) \in J\} \). This means for every element \( x \) in \( G \) that points to an element within \( J \) in the group \( H \), it is part of this preimage set.

The significance of this preimage lies in mapping relationships between two groups and understanding how substructures can be preserved under homomorphisms. It also provides insight into how subgroups are generated and maintained through various transformations.

Kernel of Homomorphism

The kernel of a homomorphism is a cornerstone concept in understanding the properties of group homomorphisms. The kernel, denoted as \( ext{ker} \, f \), is the set of all elements in \( G \) that map to the identity element in the group \( H \) when the homomorphism \( f \) is applied. Formally, \( ext{ker} \, f = \{ x \in G : f(x) = e_H \} \).

Being a kernel is significant because it helps identify the homomorphic image's structure. Importantly, the kernel of a homomorphism is always a normal subgroup of \( G \). This normal subgroup helps in partitioning \( G \) into cosets, facilitating the understanding of group structure and simplifying complex group elements into more manageable parts which are called quotient groups. Furthermore, in the context of the exercise, every element of the kernel is included within the preimage \( f^{-1}(J) \), since the identity element in \( H \) is always part of any subgroup \( J \).

Subgroup Criteria

Determining whether a set is a subgroup relies on specific criteria being met. These criteria ensure the set maintains a group structure and includes:

• Non-emptiness: The set must include at least one element. Commonly, this is shown by the presence of the identity element.
• Closure under the group operation: If any two elements \( a \) and \( b \) are in the set, then their product \( ab \) should also be in the set.
• Closure under inverses: If \( a \) is in the set, its inverse \( a^{-1} \) should also be in the set.

These three criteria ensure that the preimage of a subgroup \( f^{-1}(J) \) is indeed a subgroup of \( G \). Identifying subgroups follows a routine check of these properties, allowing group theorists to decompose groups into fundamental substructures.

Group Theory

Group theory serves as the foundational framework for studying symmetrical structures in a mathematical context. It involves the study of groups, which are sets equipped with an operation that combines any two elements to form a third, adhering to four main properties:

• Closure
• Associativity
• Existence of an identity element
• Existence of inverses for every element

Understanding how groups interact through homomorphisms and how subgroups are maintained or transformed forms a central aspect of group theory.

Homomorphisms, such as \( f: G \rightarrow H \), allow the mapping of group elements in a structured, operation-respecting way, maintaining the group's algebraic structure. Exploring these mappings provides insight into the relationships between groups and can illustrate deeper algebraic concepts, such as kernels and normal subgroups, which are essential in creating quotient groups and uncovering relationships between seemingly complex group structures.