Because A4 does not support subgroups of order 6

The alternating group A4 is a fascinating object of study in abstract algebra and group theory. It is the group of even permutations of four elements, and has a total of 12 elements. While A4 has various interesting properties, one of its distinct characteristics is the fact that it does not support subgroups of order 6. In this article, we shall explore the reasons behind this intriguing phenomenon.

Firstly, let us briefly review what a subgroup is. A subgroup of a given group G is a subset of G that is itself a group with respect to the same operation. In other words, it is a smaller group contained within the original group. Now, in order for a subgroup of order 6 to exist, it must have 6 elements and satisfy the group axioms of closure, associativity, identity, and inverse.

To explain why A4 does not support subgroups of order 6, we need to examine the structure of this group. A4 is comprised of 12 elements, which can be thought of as permutations of four distinct objects. These permutations can be visualized as rearrangements of the objects in a specific order. For instance, one element of A4 is the permutation (123), which indicates that object 1 is moved to position 2, object 2 is moved to position 3, and object 3 is moved to position 1.

It is important to note that within A4, there are no permutations that involve only three elements. In other words, any permutation in A4 must involve all four elements. This property distinguishes A4 from other permutation groups, such as the symmetric group S3, where permutations involving only three elements are possible. Since subgroups are subsets of the original group, any subgroup of order 6 of A4 would have to be constructed using permutations that involve all four elements.

However, when we consider the various permutations in A4, we find that they fall into three distinct types: identity permutations (which leave the objects unchanged), even permutations (which can be obtained by an even number of pairwise swaps), and odd permutations (which require an odd number of pairwise swaps). Furthermore, it turns out that all the permutations in A4 can be expressed as either a product of two even permutations or a product of three odd permutations. This property, known as the 3-cycle decomposition of A4, provides a key insight into why subgroups of order 6 cannot exist within A4.

To better understand this, let’s suppose that we have a subgroup H of order 6 within A4. Since subgroups must be closed under the group operation, the composition of any two elements in H should also be in H. However, due to the 3-cycle decomposition property, we would encounter a violation of closure when trying to create a subgroup of order 6. The composition of two 3-cycles in A4 would result in either a 3-cycle or a product involving four elements, neither of which would satisfy the requirement of having 6 elements.

In conclusion, the absence of subgroups of order 6 in A4 can be attributed to the unique structure and properties of this alternating group. The three distinct types of permutations in A4 and the 3-cycle decomposition property play a crucial role in preventing the formation of such subgroups. This characteristic makes A4 a fascinating subject of study for mathematicians and highlights the intricacies and beauty of abstract algebra.