Question

Without enumerating, find the conjugate subgroups of \(S_{3}\) in \(S_{4}\); of \(S_{2}\) in \(S_{4}\); of the cyclic group \([e,(123),(132)]\) in \(S_{4}\).

Short answer

The conjugate subgroups of S3 in S4 are those permuting elements (123), (124), (134), and (234). The conjugate subgroups of S2 in S4 involve the transpositions (12), (13), (14), (23), (24), and (34). The conjugate subgroups of the cyclic group [e, (123), (132)] in S4 are those generated by the permutations [e, (124), (142)], [e, (134), (143)], and [e, (234), (243)].

Step-by-step solution

Understanding Conjugate Subgroups

Conjugate subgroups in a group G are defined as two subgroups H and K for which there exists an element g in G such that gHg^{-1} = K. In the symmetric group S4, we will determine conjugates of S3, S2, and the cyclic subgroup generated by (123) and (132).

Find Conjugates of S3 in S4

Consider the subgroup S3, consisting of all permutations of three elements. We need to find all distinct ways to embed S3 in S4 by permuting its elements with an additional element. Since S4 has four elements, each conjugate of S3 will involve permutations of the three chosen elements while fixing the fourth.

Identify Conjugate Subgroups for S3 in S4

The conjugate subgroups of S3 in S4 correspond to the different choices of three elements out of four to permute -- (123), (124), (134), and (234). These are the distinct S3 subgroups in S4.

Find Conjugates of S2 in S4

S2 consists of permutations of two elements. In S4, the distinct ways to choose two elements out of four lead to six S2 subgroups, corresponding to transpositions -- (12), (13), (14), (23), (24), and (34). Each forms a conjugate subgroup of the original S2.

Find Conjugates of Cyclic Group in S4

The specified cyclic group is generated by (123), (132). To find its conjugates in S4, one must find distinct permutations of the elements 1, 2, and 3, which can be done by replacing these elements with any other trio from the set {1, 2, 3, 4}.

Identify Conjugate Subgroups for Cyclic Group in S4

For the original cyclic group, the conjugate subgroups will be generated by these permutations: [e, (124), (142)], [e, (134), (143)], and [e, (234), (243)]. These contain the same cycle structures as the original group but permute a different set of three elements.

Key concepts

Symmetric Group S4

The symmetric group S4 is an essential concept in the study of group theory, particularly in the context of permutation groups. It consists of all possible permutations of a set with four elements.

In more understandable terms, imagine having four different books. If you were to arrange these books on a shelf in every possible order, the number of different arrangements you can create is exactly what S4 represents. Mathematically, this amounts to 24 different permutations, because there are 4! (4 factorial, which equals 4×3×2×1) possibilities.

The importance of S4 lies in its role as a model for understanding how permutations work in larger sets, its applications in solving puzzles like the Rubik's Cube, and in many fields of science where the concept of symmetry is crucial. Within S4, subgroups, including smaller symmetric groups like S3 and S2, signify permutations of subsets of the four elements, and understanding the relation between these via conjugation is key for grasping the structure of permutations.

Permutation Groups

Permutation groups are mathematical structures that encapsulate the notion of shuffling objects around. Like a deck of cards that can be rearranged in many ways, a permutation group includes all the different ways to reorder a set of elements.

It is characterized by the operations of taking any two permutations and performing them successively—a process called composition—and finding the inverse of a permutation, which reverses its effect. Permutation groups are fundamental in the analysis of symmetry and are widely applicable, from puzzles and games to chemistry and physics.

In the case of the symmetric group S4, it is a permutation group for four elements, concerning every possible way to shuffle these elements. Each subgroup within S4 is a permutation group in its own right, and studying the conjugation of these subgroups helps in understanding how they interact and relate to each other within the larger group structure.

Cyclic Subgroups

Cyclic subgroups are a simpler, yet profound, type of subgroup within the broader family of permutation groups. These subgroups are generated by a single element, meaning all the permutations in the subgroup can be achieved by repeatedly applying this single permutation—a process akin to winding a clock's hands, where each hour corresponds to a different application of the same movement.

They are called 'cyclic' because this process is circular. Like a clock hitting 12 and starting over, a cyclic permutation returns to the start after a certain number of applications. In group theory, understanding cyclic subgroups offers insights into the repeating patterns within the group and helps simplify complex group structures.

For instance, within S4, a cyclic subgroup formed by permuting three elements (such as (123)) has distinct conjugates that share similar cyclic properties but involve different sets of the four elements. Each conjugate is like a variation on a theme, representing a new cycle with the same periodicity but a different starting point.