Problem 1
Question
In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. $$ \mathbb{Z}_{5} \text { and } \mathbb{Z}_{20} /\langle 5\rangle $$
Step-by-Step Solution
Verified Answer
Yes, \( \mathbb{Z}_{5} \) and \( \mathbb{Z}_{20}/\langle 5 \rangle \) are isomorphic with table elements being \([0], [1], [2], [3], [4]\) in both groups.
1Step 1: Understand the Groups
We need to show that \( \mathbb{Z}_{5} \) is isomorphic to \( \mathbb{Z}_{20}/\langle 5 \rangle \). Here, \( \mathbb{Z}_{5} \) is the group of integers modulo 5 under addition. \( \mathbb{Z}_{20} \) is the group of integers modulo 20 under addition, and \( \langle 5 \rangle \) is the subgroup generated by 5, which consists of all multiples of 5. Thus, \( \mathbb{Z}_{20}/\langle 5 \rangle \) represents the quotient group.
2Step 2: Identify the Homomorphism
Consider the natural projection homomorphism \( \phi: \mathbb{Z}_{20} \to \mathbb{Z}_{20}/\langle 5 \rangle \) defined by \( \phi(x) = x + \langle 5 \rangle \). This homomorphism maps an element \( x \) in \( \mathbb{Z}_{20} \) to its equivalence class in \( \mathbb{Z}_{20}/\langle 5 \rangle \).
3Step 3: Determine the Kernel
The kernel of \( \phi \) is the set of all integers \( x \in \mathbb{Z}_{20} \) such that \( x + \langle 5 \rangle = \langle 5 \rangle \). Hence, the kernel is \( \langle 5 \rangle \), which consists of \{0, 5, 10, 15\}.
4Step 4: Apply the Fundamental Homomorphism Theorem
The Fundamental Homomorphism Theorem states that if there is a homomorphism \( \phi: G \to H \) and \( \ker(\phi) = K \), then \( G/K \) is isomorphic to \( \phi(G) \). Here, since \( \ker(\phi) = \langle 5 \rangle \), we have \( \mathbb{Z}_{20}/\langle 5 \rangle \) is isomorphic to its image under the homomorphism.
5Step 5: Establish the Isomorphism
The image under the homomorphism is the set of equivalence classes \{[0], [1],[2], [3], [4]\}. Since this set has 5 elements, it matches the order of \( \mathbb{Z}_{5} \), which are \( \{0, 1, 2, 3, 4\} \). Thus, \( \mathbb{Z}_{20}/\langle 5 \rangle \) and \( \mathbb{Z}_{5} \) are isomorphic.
Key Concepts
Group IsomorphismKernel of a HomomorphismQuotient GroupsProjection Homomorphism
Group Isomorphism
Group isomorphism is a concept in abstract algebra that establishes a strong connection between two groups. If two groups, say \( G \) and \( H \), are isomorphic, there exists a bijective (one-to-one and onto) function \( \phi: G \to H \) that preserves the group operation. This means that for any elements \( a, b \in G \), the function satisfies \( \phi(a \cdot b) = \phi(a) \cdot \phi(b) \). Essentially, isomorphic groups have the same structure, even though the elements or the presentation of the groups might look different.
An important aspect of isomorphic groups is that they are indistinguishable in terms of their group properties. Despite possibly having different elements or operations, they behave in an identical manner, often facilitating the transfer of results or structures from one group to another.
An important aspect of isomorphic groups is that they are indistinguishable in terms of their group properties. Despite possibly having different elements or operations, they behave in an identical manner, often facilitating the transfer of results or structures from one group to another.
Kernel of a Homomorphism
The kernel of a homomorphism \( \phi: G \to H \) is a fundamental concept in group theory. It is defined as the set of elements in \( G \) that map to the identity element of \( H \). Mathematically, this can be expressed as \( \ker(\phi) = \{ g \in G \mid \phi(g) = e_H \} \), where \( e_H \) is the identity element in \( H \).
The kernel is a subgroup of the original group \( G \) and plays a crucial role in understanding the structure of the homomorphism. If the kernel is just the identity element of \( G \), the homomorphism is said to be injective, meaning \( \phi \) is a one-to-one mapping. Otherwise, the kernel provides insights into the elements of \( G \) that have been "flattened" or "identified" with the identity in \( H \), leading to the existence of possible redundancies or relations in the group structure.
The kernel is a subgroup of the original group \( G \) and plays a crucial role in understanding the structure of the homomorphism. If the kernel is just the identity element of \( G \), the homomorphism is said to be injective, meaning \( \phi \) is a one-to-one mapping. Otherwise, the kernel provides insights into the elements of \( G \) that have been "flattened" or "identified" with the identity in \( H \), leading to the existence of possible redundancies or relations in the group structure.
Quotient Groups
A quotient group, denoted as \( G/N \), is formed when a group \( G \) is partitioned by a normal subgroup \( N \). In simpler terms, the elements of \( N \) serve to group or "glue together" certain elements of \( G \) to form the elements of the new group, \( G/N \). The elements of the quotient group are the cosets of \( N \) in \( G \), and the group operation on \( G/N \) is defined in terms of these cosets.
Quotient groups are pivotal when dealing with homomorphisms and the Fundamental Homomorphism Theorem. They are used to explain how groups can "factor" through other groups, often making complex problems more approachable. In our exercise, \( \mathbb{Z}_{20}/\langle 5 \rangle \) serves as the quotient group where the elements are cosets defined by the subgroup \( \langle 5 \rangle \).
Quotient groups are pivotal when dealing with homomorphisms and the Fundamental Homomorphism Theorem. They are used to explain how groups can "factor" through other groups, often making complex problems more approachable. In our exercise, \( \mathbb{Z}_{20}/\langle 5 \rangle \) serves as the quotient group where the elements are cosets defined by the subgroup \( \langle 5 \rangle \).
Projection Homomorphism
Projection homomorphism is a specific type of homomorphism related to quotient groups. It is often used to map elements from one group to a quotient group. In our exercise, the projection homomorphism \( \phi: \mathbb{Z}_{20} \to \mathbb{Z}_{20}/\langle 5 \rangle \) is defined by \( \phi(x) = x + \langle 5 \rangle \). This mapping takes any element \( x \) in \( \mathbb{Z}_{20} \) and sends it to its corresponding coset in \( \mathbb{Z}_{20}/\langle 5 \rangle \).
Projection homomorphisms play a significant role in simplifying groups and studying their properties in a more manageable form. They are vital in the application of the Fundamental Homomorphism Theorem, ensuring that transformations between groups can preserve essential characteristics.
Projection homomorphisms play a significant role in simplifying groups and studying their properties in a more manageable form. They are vital in the application of the Fundamental Homomorphism Theorem, ensuring that transformations between groups can preserve essential characteristics.
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