Problem 16 - Algebra: Pure and Applied

Question

Let \(H\) be a subgroup of a group \(G\) and let \(X\) be the set \(\\{x H \mid x \in G\\}\) of all left cosets of \(H\) in \(G\). Let \(G\) act on \(H\) by left multiplication \((g, x H) \rightarrow g x H \in X\). Let \(\chi: G \rightarrow S_{X}\) be the permutation representation of the action. Then (a) Determine the kernel \(K\) of \(\chi\). (b) Show that \(K \subset H\) (c) Show that if \(N\) is a normal subgroup of \(G\) and \(N \subset H\), then \(N \subset K\). In other words, show that \(K\) is the largest normal subgroup of \(G\) contained in \(H\).

Step-by-Step Solution

Answer

(a) Kernel \( K \) is \( H \). (b) \( K \subset H \). (c) If \( N \) is normal and \( N \subset H \), then \( N \subset K \), making \( K \) the largest normal subgroup in \( H \).

Step 1: Define the kernel of the action

The kernel of the permutation representation \( \chi \) is defined as the set \( K = \{ g \in G \mid \chi(g) = \text{id} \} \). This means that for each element \( g \) in \( K \), the action of \( g \) on every element of the coset space \( X \) leaves it unchanged. So \( g x H = x H \) for all \( x H \in X \).

Step 2: Determine the kernel \( K \)

Given \( g x H = x H \), it implies that \( (g x) H = x H \). This means \( g x \in x H \), or equivalently, \( g x = x h \) for some \( h \in H \). This is true for all \( x \in G \), and implies that \( g \in H \). Therefore, the kernel \( K \) is such that \( K = \{ g \in G \mid g x H = x H \text{ for all } x H \in X \} = H \).

Step 3: Show that \( K \subset H \)

From Step 2, we found that \( K \subseteq H \) because for any \( g \in K \), the action of \( g \) must map all cosets \( x H \) onto themselves, implying \( g \in H \). Thus, since \( K \) is already identified as being contained within \( H \), it follows directly that \( K \subset H \).

Step 4: Show that \( N \subset K \) implies \( K \) is largest normal subgroup in \( H \)

Assume \( N \) is normal in \( G \) and \( N \subset H \). Since \( N \) is normal, for any \( g \in G \) and \( n \in N \), we have \( gng^{-1} \in N \). If \( N \subset H \), then for every element \( n \in N \), \( \chi(n) = \text{id} \) because \( n \) acts trivially on all cosets \( x H \). Thus \( N \subseteq K \). Since \( N \subset H \) and \( N \subseteq K \), \( K \) is the largest normal subgroup contained in \( H \).

Key Concepts

Subgroup

In group theory, a subgroup is a subset of a group that itself forms a group under the same operation as the larger group. Subgroups are essential as they help break down complex groups into more manageable parts. To be a subgroup, a set must satisfy three key criteria:

• Identity: The identity element of the larger group must be in the subset.
• Closure: The product of any two elements in the subset must also be in the subset.
• Inverses: The inverse of any element in the subset must also be in the subset.

By using these rules, we can identify and verify any subgroup within a group. This concept is particularly important because it allows us to explore properties of the larger group by examining its smaller constituent parts.

Cosets

Cosets are used to structure groups into simpler forms and are pivotal in the study of groups. Given a subgroup \(H\) within a group \(G\), a coset is formed by multiplying every element of \(H\) with a fixed element from \(G\). There are two types of cosets:

• Left Cosets: These take the form \(gH = \{gh \mid h \in H\}\).
• Right Cosets: These are structured as \(Hg = \{hg \mid h \in H\}\).

Cosets help in partitioning the group \(G\) into equal-sized, non-overlapping parts, with each partition size being equal to the size of \(H\). This partitioning is useful for understanding the structure of \(G\) and leads directly to the concept of a quotient group, which provides insights into the larger group's properties.

Normal Subgroup

A normal subgroup is a special type of subgroup that's invariant under conjugation by any element of the group. That is, for a subgroup \(N\) to be considered normal in a group \(G\), it must satisfy the condition \(gNg^{-1} = N\) for all \(g \in G\). Normal subgroups are crucial because they allow us to form quotient groups simply and elegantly, which in turn aids in understanding complex group structures.

• Normal subgroups are essential in the study of group homomorphisms, as the kernel of a homomorphism is always a normal subgroup.
• They serve as the building blocks for constructing simple and factor groups.

Given the importance of normal subgroups, identifying them often helps in assessing whether a group can be broken down into simpler or more basic components.

Permutation Representation

Permutation representation is a way of expressing abstract groups as groups of permutations, which are bijective functions from a set onto itself. This concept is valuable because it allows abstract group structures to be represented concretely by permutations, making them easier to visualize and analyze.

• A permutation representation of a group \(G\) involves a homomorphism from \(G\) to the symmetric group of permutations on a certain set.
• It is particularly used to study the ways in which a group can act on a set.

The action of \(G\) on a set can be tracked through these permutations, revealing insights about not just what elements are present, but how they can transform into each other under the group's operation. This representation is instrumental in simplifying and understanding the behavior and symmetry of groups in areas like algebra and geometry.