Proving Aut(V4) is Isomorphic to S3: A Comprehensive Guide

The automorphism group of the Klein 4-group, denoted as Aut(V4), is a fascinating topic in group theory. In this article, we will walk through the process of proving that Aut(V4) is isomorphic to the symmetric group S3. This proof will involve several steps, including identifying the structure of the Klein 4-group, determining its automorphisms, and finally verifying the isomorphism.

Understanding the Klein 4-Group (V4)

The Klein 4-group, denoted as V4, is a group with four elements that is isomorphic to the direct product of two cyclic groups of order 2. It can be represented as follows:

V4 { e, a, b, c }

The multiplication rules for V4 are:

idementity element e a2 b2 c2 e ab c, ac b, bc a

It is an abelian group, meaning the group operation is commutative.

Determining the Automorphisms of V4

An automorphism of a group is a bijective homomorphism from the group to itself. To find the automorphisms of V4, we need to determine how the non-identity elements (a, b, c) can be mapped to other non-identity elements while preserving the group structure.

Since V4 is abelian, any automorphism can be determined by the images of its generators. The non-identity elements are a, b, c, and they can be mapped to each other in such a way that the group structure is preserved.

Counting the Automorphisms

Each non-identity element must map to another non-identity element, and there are three non-identity elements: a, b, c. The number of ways to map these elements is as follows:

Pick one non-identity element, say a, which can be mapped in 3 ways. The second non-identity element, say b, can be mapped in 2 ways after the first choice is made. The third non-identity element, c, will be determined by the first two choices.

Thus, the total number of ways to map the three non-identity elements is:

3! 6

Identifying the Structure of Aut(V4)

The group of all permutations of the three non-identity elements {a, b, c} corresponds to the symmetric group S3, which has 6 elements. Therefore, we have:

Aut(V4) ? S3

Conclusion and Verification

To complete the proof, we verify that Aut(V4) is a non-commutative group of order 6. This is done by considering two automorphisms: a cyclic map u that permutes a, b, c cyclically, and a map v that interchanges b and c, leaving e and a fixed. When we compute the compositions uv and vu, they are different, which confirms that Aut(V4) is non-commutative.

Since Aut(V4) is a non-commutative group of order 6, it must be isomorphic to the symmetric group S3. Thus, we conclude: