Problem 2
Question
Let \(H\) and \(K\) be normal subgroups of a group \(G\), with \(H \subseteq K\). Define \(\phi: G / H \rightarrow G / K\) by: \(\phi(H a)=K a\). Prove the following : $$ \phi \text { is a homomorphism. } $$
Step-by-Step Solution
Verified Answer
The map \( \phi \) is a homomorphism because \( \phi(Ha \cdot Hb) = \phi(Ha) \cdot \phi(Hb) \) holds for all elements in \( G/H \).
1Step 1: Define the map and input elements
We are given the map \( \phi: G/H \to G/K \) defined by \( \phi(Ha) = Ka \). We need to check if this map is a homomorphism by taking two arbitrary elements \( Ha, Hb \in G/H \) and showing that \( \phi(Ha \cdot Hb) = \phi(Ha) \cdot \phi(Hb) \).
2Step 2: Compute the left side of the homomorphism definition
First, compute \( Ha \cdot Hb \) in the quotient group \( G/H \). Recall that \( Ha \cdot Hb = H(ab) \). Apply \( \phi \) to this, giving \( \phi(H(ab)) = K(ab) \).
3Step 3: Compute the right side of the homomorphism definition
Next, apply \( \phi \) to \( Ha \) and \( Hb \) individually: \( \phi(Ha) = Ka \) and \( \phi(Hb) = Kb \). Compute the product in \( G/K \): \( (Ka)(Kb) = K(ab) \) due to the properties of the quotient group where multiplication is defined component-wise by the cosets.
4Step 4: Conclude the homomorphism proof
We have shown that \( \phi(H(ab)) = K(ab) \) and also \( \phi(Ha) \cdot \phi(Hb) = K(ab) \). Since both expressions are equal in \( G/K \), we conclude that \( \phi(Ha \cdot Hb) = \phi(Ha) \cdot \phi(Hb) \). Therefore, \( \phi \) is a homomorphism.
Key Concepts
Group HomomorphismQuotient GroupCosets
Group Homomorphism
A group homomorphism is a beautiful structure-preserving map between two groups. It allows us to explore the intricate dance between algebraic operations in different contexts. In this exercise, we define a map \( \phi: G/H \rightarrow G/K \). Our task is to determine if this map \( \phi \) is a homomorphism.
Simply put, a map is a homomorphism if it respects the group operation. For our case, this means that for any elements \( Ha, Hb \) in the quotient group \( G/H \), the image of their product must equal the product of their images. Mathematically, this is written as \( \phi(Ha \cdot Hb) = \phi(Ha) \cdot \phi(Hb) \).
By analyzing the original exercise and the provided solution, we apply the function to both sides of the operation. We find that \( \phi(H(ab)) = K(ab) \) aligns with \( (Ka)(Kb) = K(ab) \). Thus, our function respects the structure of the groups involved, verifying that \( \phi \) is indeed a homomorphism.
Simply put, a map is a homomorphism if it respects the group operation. For our case, this means that for any elements \( Ha, Hb \) in the quotient group \( G/H \), the image of their product must equal the product of their images. Mathematically, this is written as \( \phi(Ha \cdot Hb) = \phi(Ha) \cdot \phi(Hb) \).
By analyzing the original exercise and the provided solution, we apply the function to both sides of the operation. We find that \( \phi(H(ab)) = K(ab) \) aligns with \( (Ka)(Kb) = K(ab) \). Thus, our function respects the structure of the groups involved, verifying that \( \phi \) is indeed a homomorphism.
Quotient Group
The concept of a quotient group is fundamental in group theory because it helps us understand deep aspects about groups by considering their "coset layers."
Given a group \( G \) and a normal subgroup \( H \), the quotient group \( G/H \) is formed by the cosets of \( H \) in \( G \). In other words, in the group \( G/H \), each element can be thought of as a collection of elements from \( G \) grouped together by their relationship to \( H \).
For example, if you have a normal subgroup \( K \) like in our exercise, \( G/K \) would be seen as the group of cosets of \( K \) in \( G \). The operation defined on the quotient group is component-wise multiplication of these cosets. This means we multiply two elements from \( G/H \) or \( G/K \) by multiplying representatives from each coset and taking their coset in the respective normal subgroup.
Quotient groups simplify complex group structures, and in this problem, we witness their elegance in action with the map \( \phi \), which preserves the operation from \( G/H \) to \( G/K \).
Given a group \( G \) and a normal subgroup \( H \), the quotient group \( G/H \) is formed by the cosets of \( H \) in \( G \). In other words, in the group \( G/H \), each element can be thought of as a collection of elements from \( G \) grouped together by their relationship to \( H \).
For example, if you have a normal subgroup \( K \) like in our exercise, \( G/K \) would be seen as the group of cosets of \( K \) in \( G \). The operation defined on the quotient group is component-wise multiplication of these cosets. This means we multiply two elements from \( G/H \) or \( G/K \) by multiplying representatives from each coset and taking their coset in the respective normal subgroup.
Quotient groups simplify complex group structures, and in this problem, we witness their elegance in action with the map \( \phi \), which preserves the operation from \( G/H \) to \( G/K \).
Cosets
Cosets are crucial in understanding the structure of groups and how they interact with subgroup elements. When we talk about cosets, we're referring to subsets formed by taking a subgroup and shifting it by elements of the larger group.
To visualize, consider a group \( G \) and a subgroup \( H \). Each coset in \( G \) of \( H \) takes the form \( aH \) where \( a \) belongs to \( G \). This creates a "shifted" version of \( H \) in \( G \), and all the elements of this coset are simply multiples of the form \( ah \) for every \( h \) in \( H \).
In our exercise, cosets are used to explore the group relations within the quotient groups \( G/H \) and \( G/K \). The map \( \phi(Ha) = Ka \) demonstrates how cosets in \( G/H \) translate to cosets in \( G/K \). Each time you move from an expression like \( Ha \) to \( Ka \), you're highlighting how the structures remain consistent even as the group morphs through this mapping.
Ultimately, cosets take something potentially overwhelming like a group and break it into more manageable pieces. They allow us to understand the underlying algebraic tapestry and to appreciate their role in transformations like group homomorphisms.
To visualize, consider a group \( G \) and a subgroup \( H \). Each coset in \( G \) of \( H \) takes the form \( aH \) where \( a \) belongs to \( G \). This creates a "shifted" version of \( H \) in \( G \), and all the elements of this coset are simply multiples of the form \( ah \) for every \( h \) in \( H \).
In our exercise, cosets are used to explore the group relations within the quotient groups \( G/H \) and \( G/K \). The map \( \phi(Ha) = Ka \) demonstrates how cosets in \( G/H \) translate to cosets in \( G/K \). Each time you move from an expression like \( Ha \) to \( Ka \), you're highlighting how the structures remain consistent even as the group morphs through this mapping.
Ultimately, cosets take something potentially overwhelming like a group and break it into more manageable pieces. They allow us to understand the underlying algebraic tapestry and to appreciate their role in transformations like group homomorphisms.
Other exercises in this chapter
Problem 2
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