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Magazine Chapter 9: Problem 9
Bestimmen Sie alle Automorphismen der symmetrischen Gruppe \(S_{3}\).
Short Answer
Expert verified
All automorphisms of \( S_3 \) are inner automorphisms.
Step by step solution
01
Understanding Automorphisms
An automorphism of a group is a bijective homomorphism from the group to itself. Since we are dealing with the symmetric group \( S_3 \), our goal is to find all permutations \( \varphi : S_3 \rightarrow S_3 \) such that \( \varphi(ab) = \varphi(a)\varphi(b) \) for all \( a, b \in S_3 \) and \( \varphi \) is bijective.
02
Structure of the Symmetric Group
The symmetric group \(S_3\) consists of 6 elements: the identity \(e\), three transpositions \((12), (13), (23)\), and two 3-cycles \((123), (132)\). It is a non-Abelian group, which means the order in which operations are performed matters.
03
Identifying Inner Automorphisms
By Cayley's theorem, every group is isomorphic to a subgroup of its symmetric group. Inner automorphisms can be represented by \( \varphi_g(x) = gxg^{-1} \) for some fixed \( g \in S_3 \). Calculate \( \varphi_g \) for each element \( g \in S_3 \) and verify these are all automorphisms.
04
Computing Inner Automorphisms
For any \( g \in S_3 \), compute \( \varphi_g(x) = gxg^{-1} \) for each element \( x \in S_3 \). Verify that this transformation is a homomorphism and that all such transformations produced are distinct, mapping onto the group itself.
05
Determine Additional Automorphisms
For symmetric groups \( S_n \) where \( n eq 6 \), all automorphisms are inner. Check that all automorphisms found from inner transformations exhaust the set of automorphisms for \( S_3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetric Group
The symmetric group, denoted as \( S_n \), is a key concept in group theory. It consists of all the permutations of \( n \) elements, and its order is \( n! \), which represents the total number of these permutations. In essence, \( S_n \) is the group of all bijections from a set with \( n \) elements to itself. For example, the symmetric group \( S_3 \) includes all the possible ways to arrange three different objects.
- Elements: The elements of \( S_3 \) include the identity permutation \( e \), three transpositions \((12), (13), (23)\), and two 3-cycles \((123), (132)\).
- Properties: The group \( S_3 \) is non-Abelian, meaning that the order of applying group elements matters. For example, applying the cycle \((123)\) followed by \((13)\) will yield a different result than applying \((13)\) first.
Cayley's Theorem
Cayley's Theorem is a fundamental statement in group theory that shows any group is, in fact, equivalent to a group of permutations. Essentially, it states that every group \( G \) is isomorphic to a subgroup of the symmetric group acting on \( G \). This means that any group can be visualized as permutations of some set, reaffirming the prominence of permutation groups in group theory.
- Isomorphisms: Because of Cayley's Theorem, understanding permutations helps in understanding any abstract group since it can be represented as a permutation group.
- Practical Implication: For example, considering \( S_3 \), the theorem implies that it can represent all possible automorphisms of a set with three elements by permutations.
Non-Abelian Group
A non-Abelian group is defined by its lack of commutativity; in other words, for some elements \( a \) and \( b \) in a group \( G \), \( ab eq ba \). This characteristic makes non-Abelian groups richer in structure and interesting to study.
- Examples: The smallest example of a non-Abelian group is \( S_3 \), the symmetric group on three elements as the order of operations affects the outcome.
- Significance: Non-Abelian groups are crucial for understanding more complex algebraic structures, as most interesting algebraic settings are non-Abelian.
Inner Automorphisms
Inner automorphisms are a specific type of group automorphism that arise from conjugation. In the context of a group \( G \), an inner automorphism is represented by \( \varphi_g(x) = gxg^{-1} \), where \( g \) is a fixed element of \( G \). Inner automorphisms form a subgroup of the automorphism group of \( G \).
- Definition: Inner automorphisms are transformations of a group that maintain its structure, mapping elements to their conjugates.
- Connection to \( S_3 \): Within \( S_3 \), computing inner automorphisms requires finding all possible configurations \( gxg^{-1} \) for a given \( g \), ensuring these represent valid automorphisms of the group.
