Question

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

Short answer

\(\operatorname{Aut}(S_3) \cong S_3\) because automorphisms permute its elements similar to their permutations in \(S_3\).

Step-by-step solution

Identify Automorphisms

The automorphism group, \(\operatorname{Aut}(S_3)\), consists of all bijective homomorphisms from the symmetric group \(S_3\) to itself. Remember, if \(\phi\) is an automorphism of \(S_3\), it must permute all elements while preserving group structure.

Analyze Group Structure of \(S_3\)

The symmetric group \(S_3\) is the group of all permutations of three elements, with 6 elements in total. These elements can be characterized by the cycle structure: there are 3 transpositions \((1\,2), (2\,3), (1\,3)\), one identity element \((\))/e, and two 3-cycles \((1\,2\,3)\) and \((1\,3\,2)\).

Determine Automorphism Properties

Automorphisms must map elements with the same cycle type to each other. Thus, all transpositions can be mapped to each other, and similarly, 3-cycles are mapped among themselves. This mapping is essentially another permutation of the elements of \(S_3\).

Find Generating Set for \(S_3\)

The group \(S_3\) can be generated using two elements, e.g., the transpositions \((1\,2)\) and \((1\,3)\). Any automorphism is determined entirely by its action on these generators.

Establish a Correspondence

Since any automorphism is determined by its action on the generators, note that a bijective map between sets of generators defines a homomorphism. Here, any reordering of transpositions and 3-cycles corresponds to a permutation of the generating set of 6 elements of \(S_3\) itself.

Conclude Isomorphism

The key idea is that each automorphism corresponds to a rearrangement of the elements inside \(S_3\) that respects the group's operation. Therefore, \(\operatorname{Aut}(S_3)\) is isomorphic to \(S_3\) itself. Thus, \(\operatorname{Aut}(S_3) \cong S_3\).

Key concepts

Symmetric Group

The concept of a symmetric group is foundational in group theory. Specifically, the symmetric group of degree 3, denoted as \( S_3 \), represents all possible permutations of three distinct elements. The idea is simple: think of \( S_3 \) as all the ways you can rearrange three unique items. This group is made up of 6 elements because there are 6 possible ways to order those items. In combination notation, this can be expressed as 3 factorial (3!), which equals 6.

• There are three transpositions: single swaps between pairs of elements. For instance, swapping position 1 with 2 or 2 with 3.
• The identity element, which leaves all positions unchanged.
• Two 3-cycles, which rotate all three elements consecutively.

Understanding \( S_3 \) is crucial because it serves as a prototype for the symmetric groups in general, illustrating fundamental properties like closure, identity, inverses, and associativity, which are inherent in all groups.

Automorphism

An automorphism in group theory is essentially a transformation of a group onto itself. For the symmetric group \( S_3 \), an automorphism rearranges the elements while preserving their structure and interactions. Think of it as reshuffling the elements but keeping the core 'rules' of the group intact.
These mappings are bijections, meaning they pair each element with a unique image, without overlaps or omissions. What's fascinating is that any automorphism must map like cycle structures to each other—for example, transpositions to transpositions, and 3-cycles to 3-cycles.

• All possible automorphisms of \( S_3 \) can be seen as permutations.
• The identity automorphism does nothing, leaving \( S_3 \) unchanged.
• Each automorphism is actually a way to reassociate the generation of \( S_3 \).

Thus, exploring automorphisms in \( S_3 \) helps uncover deeper symmetries and invariances within the group.

Group Isomorphism

The term 'group isomorphism' denotes a deep equivalence between two groups, meaning they have the same structural properties and operational forms, even if their elements might appear different. The notation \( \operatorname{Aut}(S_3) \cong S_3 \) indicates such a relationship, signifying that the automorphism group of \( S_3 \) is isomorphic to \( S_3 \) itself.
To establish a group isomorphism, a map must meet certain criteria:

• Both injective (no duplicates in mapping) and surjective (covers all elements).
• Operation preserving: the product of the image must correspond to the image of the product.

In simpler terms, each group operation in one corresponds exactly to operations in the other, maintaining the structure and relationships intact. Through isomorphism, groups can be seen as essentially the same concerning their intrinsic properties, even when superficially different.

Cycle Structure

Cycle structure in group theory refers to the unique arrangement and interaction of a group's elements. For the symmetric group \( S_3 \), the cycle structure is integral to understanding how the group functions and interacts.

• A cycle indicates a repeating transition path among elements, such as \((1\,2\,3)\), which means 1 goes to 2, 2 goes to 3, and 3 goes back to 1.
• Cycles can be combined to create more complex permutations through multiplication.
• Understanding cycles also aids in comprehending the essence of automorphisms, which map elements based on their cycle forms.

Cycles simplify the observation of permutations, making it easier to predict outcomes and recognize patterns within the group. Ultimately, the cycle structures inform how automorphisms preserve symmetries and how isomorphisms between groups can be verified by matching these structural signatures.