Problem 6 - A Course In Group Theory

Question

Let \(G\) be the group \(S(3) .\) Calculate \(N_{G}(H)\) when \(H\) is (a) the subgroup \(\\{1,(123),(132)\\}\) and (b) the subgroup \(\\{1,(12)\\}\).

Step-by-Step Solution

Answer

(a) \(N_G(H) = S_3\), (b) \(N_G(H) = \{1, (12)\}\).

Step 1: Understanding Normalizers

The normalizer of a subgroup \(H\) in a group \(G\), denoted \(N_G(H)\), is the set \(\{ g \in G \mid gHg^{-1} = H \}\). This means that for each \(g\) in the normalizer, conjugation of \(H\) by \(g\) leaves \(H\) unchanged.

Step 2: Analyze Subgroup (a)

For ans (a), consider subgroup \(H = \{1, (123), (132)\}\). Notice that \(H\) itself is a normal subgroup of \(S_3\) since it contains all of its elements' permutations forming a group. Hence, the normalizer \(N_{S_3}(H)\) is the group \(S_3\) itself, as conjugating \(H\) by any element in \(G\) leaves \(H\) unchanged because \(H\) is normal.

Step 3: Analyze Subgroup (b)

For ans (b), consider subgroup \(H = \{1, (12)\}\). Check which elements of \(S_3\) conjugate \((12)\) to itself. Calculate: \((13)(12)(13) = (23)\), \((23)(12)(23) = (13)\). Only conjugation by elements \(1\) and \((12)\) leave \((12)\) invariant. So, \(N_{S_3}(H) = \{ 1, (12) \}\), leaving the subgroup unchanged.

Key Concepts

Normalizer of a Subgroup

When studying group theory, one interesting concept is the normalizer of a subgroup. Imagine you have a subgroup, let's call it \( H \), within a larger group \( G \). The normalizer, denoted \( N_G(H) \), consists of all elements in \( G \) that, when you perform a specific operation called conjugation on \( H \), leaves \( H \) unchanged.

In math terms, for each element \( g \in G \), we say \( gHg^{-1} = H \) if \( g \) is in the normalizer. Conjugating \( H \) by \( g \) means multiplying each element of \( H \) by \( g \), then the inverse of \( g \). If the set of elements output from this operation is exactly \( H \) again, \( g \) is part of \( N_G(H) \).

• Acts as a stabilizer for \( H \)
• Shows symmetry within \( G \)

The normalizer helps us understand how 'self-contained' or 'stable' a subgroup is within a larger group.

Symmetric Group

A symmetric group is a concept from the field of abstract algebra, specifically group theory, and is usually denoted by \( S_n \). It is the group of all possible permutations, or rearrangements, of \( n \) elements. Think of it as all possible ways to shuffle a deck of \( n \) cards.

For example, \( S_3 \) is the symmetric group on three elements, meaning it includes all permutations of the set \( \{1, 2, 3\} \). This group contains six permutations because there are six possible ways to arrange three elements:

• Identity: \( (1) \)
• Transpose two elements: \( (12), (13), (23) \)
• Cycles: \( (123), (132) \)

Symmetric groups are pivotal in understanding many structures in mathematics because they serve as key examples of permutation groups, which are fundamental in studying symmetries.

Conjugation in Groups

Conjugation is a central operation in group theory and involves transforming an element within the group using another element from the group. Conjugation is denoted as \( gHg^{-1} \), where \( g \) is an element of the group \( G \) and \( H \) is a subgroup.

Here's how it works: for a given element \( h \in H \), you multiply it by a group element \( g \), then multiply the result by the inverse of \( g \). This operation is symbolic of symmetry and often shows how elements and substructures are related within a group.

• Illustrates how subgroups relate to each other under group actions
• Can show if a subgroup is invariant (or normal) under conjugation

Understanding conjugation helps identify which elements in the group "play nice" with elements in the subgroup or with the subgroup itself.

Normal Subgroups

A normal subgroup is a special type of subgroup within a larger group where every element of the group \( G \) conjugates the subgroup into itself. In simpler terms, a subgroup \( H \) is normal in \( G \), denoted by \( H \trianglelefteq G \), if for every element \( g \in G \), the operation \( gHg^{-1} = H \) holds.

One way to think about normal subgroups is that they are "well-behaved" under the group operations. They are important because they allow the group to be divided into simpler pieces—called factor groups or quotient groups—which help in understanding the group's overall structure.

• Key in studying group homomorphisms
• Important for constructing new groups through quotient groups

Understanding normal subgroups improves the comprehension of group theory's complex structure and how large groups can be built from smaller building blocks.