Problem 22

Question

Find the normalizer of the indicated subgroup in the indicated group. $$ A_{3} \text { in } S_{3} $$

Step-by-Step Solution

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Answer

The normalizer of $ A_3 $ in $ S_3 $ is $ S_3 $.

1Step 1: Understand the Group and Subgroup

First, we need to recall that the group $ S_3 $ is the symmetric group of degree 3, which consists of all permutations of three objects. $ S_3 $ has order 6 and contains 6 elements: the identity $ (1) $, the 3 transpositions $ (12), (13), (23) $, and the 2 cyclic permutations $ (123), (132) $. The subgroup $ A_3 $ is the alternating group of degree 3, which is the subgroup of even permutations in $ S_3 $ and contains 3 elements: $ (1), (123), (132) $.

2Step 2: Define the Normalizer

The normalizer of a subgroup $ H $ in $ G $, denoted $ N_G(H) $, is defined as the set of all elements $ g $ in $ G $ such that $ gH = Hg $. In other words, the normalizer is the largest subgroup of $ G $ in which $ H $ is normal.

3Step 3: Calculate Conjugates

To find the normalizer, we calculate all conjugates of elements of $ A_3 $ in $ S_3 $. For every $ g \in S_3 $ and every $ h \in A_3 $, calculate $ ghg^{-1} $. Since $ A_3 $ is a subgroup of order 3 and $ S_3 $ is the full symmetric group, it is straightforward to check that for all elements of $ S_3 $, conjugation within $ A_3 $ just rearranges the elements of $ A_3 $, showing that every element of $ S_3 $ conjugates $ A_3 $ to itself.

4Step 4: Conclusion for the Normalizer

Since any element of $ S_3 $ conjugates $ A_3 $ into itself, $ A_3 $ is normal in $ S_3 $. This implies that the normalizer of $ A_3 $ in $ S_3 $ is $ S_3 $ itself. Thus, $ N_{S_3}(A_3) = S_3 $.

Key Concepts

symmetric groupalternating groupconjugate elementsgroup normalizer

symmetric group

A symmetric group, commonly denoted as $ S_n $, is a fundamental concept in group theory and abstract algebra. It represents all possible permutations of $ n $ objects. For example, the symmetric group $ S_3 $ deals with permutations of three elements, say $ \{1, 2, 3\} $. Each permutation rearranges these elements in a different order.

Key Properties of Symmetric Group:

Order: The order of a symmetric group $ S_n $ is $ n! $. So, $ S_3 $ has an order of 6, because there are 6 possible ways to arrange three elements.
Elements: It includes all possible permutations, which consist of the identity permutation and transpositions (swaps of two elements).
Structure: Symmetric groups are not commutative, meaning the order of applying permutations matters.

Understanding the symmetric group is fundamental for studying more complex group structures, like alternating groups, and for solving problems involving permutations.

alternating group

The alternating group, denoted $ A_n $, is a special subgroup of the symmetric group $ S_n $. It consists only of the even permutations, which are permutations that can be achieved by an even number of transpositions.

Characteristics of Alternating Group:

Order: The order of $ A_n $ is $ \frac{n!}{2} $. For instance, $ A_3 $ contains 3 elements because 3! is 6 and dividing by 2 gives us 3.
Elements: In $ A_3 $, the elements are $ (1) $, $ (123) $, and $ (132) $.
Importance: Alternating groups are important in the study of permutation groups because $ A_n $ for $ n \geq 5 $ is a simple group, meaning it does not have any normal subgroups other than the trivial group and itself.

Studying alternating groups helps in understanding how permutations function under restrictions, such as the condition of being 'even'.

conjugate elements

In group theory, two elements $ a $ and $ b $ of a group $ G $ are called conjugate if there exists an element $ g $ in $ G $ such that $ b = gag^{-1} $. Conjugate elements share many properties and often behave similarly in group operations.

Features of Conjugate Elements:

Relation: Conjugacy is an equivalence relation among the group elements, meaning it is reflexive, symmetric, and transitive.
Class: The set of all elements conjugate to a given element forms a conjugacy class.
Utility in Normalizers: Understanding conjugacy is vital when determining normalizers, as it helps in verifying which elements can transform a subgroup into itself.

Conjugate elements provide insight into the internal structure of a group, identifying how elements interact and transform under group operations.

group normalizer

The concept of a group normalizer is critical when understanding how subgroups integrate within larger groups. The normalizer $ N_G(H) $ of a subgroup $ H $ in a group $ G $ is the largest subgroup in which $ H $ is normal. Basically, it's the set of all elements $ g $ in $ G $ that conform to the condition $ gH = Hg $.

Key Aspects of Group Normalizers:

Stability: The normalizer helps determine the stability of $ H $ within $ G $. It indicates how $ H $ behaves under conjugation by elements of $ G $.
Significance: If $ H $ is normal in $ G $, then the normalizer is $ G $ itself.
Application: Normalizers are used in the calculation of centralizers, transporters, and when determining if a subgroup is a fully invariant subgroup.

The idea of normalizers assists in exploring the symmetries within mathematical structures, particularly when analyzing subgroup behavior under group actions.

Problem 21

Problem 22

Other exercises in this chapter

Problem 21

Show that $\operatorname{Aut}\left(Q_{8}\right) \cong S_{4}$

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Problem 21

Let $G$ be a group with $|G|=p q,$ where $p$ and $q$ are primes. Show that every proper subgroup of $G$ is cyclic.

View solution

Problem 22

Consider the dihedral group \(D_{n}=\left\\{\rho^{i} \tau^{j} \mid 0 \leq i

View solution

Problem 22

Show that $\operatorname{Aut}\left(S_{3}\right) \cong S_{3}$

View solution