Symmetric Group S_4 and Its Subgroups of Order 4 and 6

The symmetric group S_4 is a fundamental object in group theory, representing all permutations of four elements. It is well known for its rich structure and diverse subgroups. This article explores the existence of subgroups of orders 4 and 6 within S_4, with a focus on their constructions and properties.

Introduction to Symmetric Groups and Subgroups

The symmetric group S_n consists of all permutations of n elements. For (n 4), this group, denoted as S_4, includes all possible permutations of four elements. The order of S_4 is 24, and it has several interesting subgroups of specific orders, including those of order 4 and 6. This article provides a comprehensive analysis of such subgroups within S_4.

Subgroups of Order 4 in S_4

To find subgroups of order 4 within S_4, we explore two primary types of subgroups:

Klein Four-Group (V_4): The Klein four-group is a well-known abelian group that can be represented by the subgroup consisting of the identity and three double transpositions. Specifically, V_4 can be described as the set of permutations: [ V_4 { (1234), (1324), (1423) }. ] Cyclic Group Generated by 4-Cycles: A cyclic subgroup of order 4 can be generated by a 4-cycle, such as (1234). This cyclic subgroup is denoted as ([1234] { (1234), (1324), (1432) }).

The analysis of subgroups of order 4 in S_4 is supported by computational evidence from the GAP system, as shown below:

CJ:ConjugacyClassesSubgroupsSymmetricGroup4

ListCJ x - OrderRepresentative(x)

[ 1, 2, 2, 3, 4, 4, 4, 6, 8, 12, 24 ]

This output confirms the presence of multiple subgroups of various orders, including orders 4 and 6.

Subgroups of Order 6 in S_4

Subgroups of order 6 in S_4 can be constructed by considering subgroups isomorphic to S_3, which can be embedded within S_4. One common subgroup is generated by the permutations (123) and (12). Specifically, the subgroup generated by (123) and (12) is isomorphic to S_3 and consists of the following permutations:

(123) (132) (12) (13) (23)

It is important to note that there are four such subgroups in S_4, each fixing one of the four elements in the symmetric group, as shown below:

Cyclic and Symmetric Group Subgroups: For S_4, the subgroups isomorphic to S_3 can be constructed by fixing one of the four elements. This results in four distinct subgroups of order 6, each isomorphic to S_3.

Summary and Further Exploration

In summary:

Subgroups of order 4: S_4 has subgroups of order 4, specifically the Klein four-group and cyclic groups generated by 4-cycles. Subgroups of order 6: S_4 contains subgroups of order 6, each isomorphic to S_3.

The total number of subgroups of order 4 can be further refined. A non-cyclic subgroup of order 4 in S_4 would be isomorphic to (mathbb{Z}_2 oplus mathbb{Z}_2), which implies all elements must be of order 2. This type of subgroup comprises permutations that are either transpositions or products of disjoint transpositions. Computationally, there are exactly seven subgroups of order 4 in S_4(), specifically one normal subgroup and three non-normal subgroups along with three cyclic subgroups.

Conclusion

The rich structure of the symmetric group S_4 allows for a diverse array of subgroups. By exploring subgroups of order 4 and 6, we gain deeper insights into the mathematical properties and group theory concepts. Understanding these subgroups is crucial in various applications of group theory in mathematics and related fields.