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Magazine Chapter 2: Problem 22
Find the normalizer of the indicated subgroup in the indicated group. $$ A_{3} \text { in } S_{3} $$
Short Answer
Expert verified
The normalizer of \( A_3 \) in \( S_3 \) is \( S_3 \).
Step by step solution
01
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: \( (1), (123), (132) \).
02
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.
03
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.
04
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, \( N_{S_3}(A_3) = S_3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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:
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.
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:
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.
