Question

Find four different subgroups of \(S_{4}\) that are isomorphic to \(S_{3}\).

Short answer

There are four subgroups isomorphic to \(S_3\) in \(S_4\), each formed by permutations of three out of four elements.

Step-by-step solution

Understand the Properties of Groups

The symmetric group on four objects, denoted as \(S_4\), consists of all permutations of four elements and has an order of 24. The symmetric group on three objects, \(S_3\), consists of all permutations of three elements and has an order of 6. Our task is to find subgroups within \(S_4\) that have the same structure as \(S_3\), i.e., they must also have an order of 6.

Identifying Possible Subgroups

A subgroup is isomorphic to \(S_3\) if it is also a permutation group on three objects. In \(S_4\), we can choose any three elements out of the four to form a subgroup whose structure matches \(S_3\). Each choice of three elements from four can form a subgroup that consists of all permutations of those three elements.

Using Combinatorial Selection

To choose three elements from a set of four elements, we use combinations. The number of ways to choose three objects from four is given by the combination formula \( \binom{4}{3} = 4 \). Hence, there are four distinct ways to choose three elements, leading to four different subgroups isomorphic to \(S_3\).

Construct Specific Subgroups

The four distinct subgroups are formed as follows:- Subgroup formed by permutations of \( \{1,2,3\} \).- Subgroup formed by permutations of \( \{1,2,4\} \).- Subgroup formed by permutations of \( \{1,3,4\} \).- Subgroup formed by permutations of \( \{2,3,4\} \).Each subgroup consists of the identity element and all possible permutations of their chosen three elements.

Key concepts

Isomorphic Groups

In group theory, two groups are called isomorphic if there is a one-to-one correspondence between their elements that preserves the group operation. Essentially, isomorphic groups are structurally identical, even if they might appear different at first glance. This means that there's a way to match elements from one group to the elements of the other, such that the operation (like addition or multiplication) behaves in the same way in both groups.
An isomorphism between two groups allows us to treat them as the same from a structural viewpoint. Thus, when we say a subgroup of the symmetric group, denoted as \(S_4\), is isomorphic to \(S_3\), we mean that there's a way to rearrange its elements such that the permutations have the same relational structure as those in \(S_3\).
This concept is fundamentally important in abstract algebra since it allows us to classify groups based on their structure rather than their representation.

Permutation Group

A permutation group is a mathematical concept wherein the elements of the group represent permutations of a given set. Permutations are essentially rearrangements of a set’s elements. In the context of symmetric groups such as \(S_3\) and \(S_4\), we're talking about permuting three and four elements, respectively.
In \(S_4\), the permutations include every possible way to order four distinct elements, which results in 24 permutations in total. Similarly, \(S_3\) involves ordering three elements, resulting in 6 permutations. These permutations form a group because they satisfy the group axioms:

• Closure: Performing any number of permutations is also a permutation.
• Associativity: The order of applying permutations doesn't change the outcome.
• Identity: There exists a permutation that leaves all elements unchanged.
• Inverse: Each permutation can be reversed with another permutation.

Understanding permutation groups helps to appreciate how groups can organize and reorder elements within a given set.

Group Order

The order of a group refers to the total number of its elements. This is an important concept because it provides insight into the complexity of the group. For the symmetric group \(S_4\), the order is 24 because it consists of all possible permutations of a four-element set.
In contrast, \(S_3\) has an order of 6, as it includes all permutations of a three-element set. When finding subgroups of \(S_4\) that are isomorphic to \(S_3\), we are essentially looking for subsets of \(S_4\) that have the same order, which in this case is 6.
It's crucial to recognize the role of group order in the classification of groups, as it often dictates the group's possible structures and behaviors.

Combinatorial Selection

Combinatorial selection involves choosing subsets from a larger set and is often represented using combinations. The concept is key when identifying subgroups isomorphic to a smaller group within a larger one, such as finding \(S_3\) structures within \(S_4\).
The combinatorial problem here is to choose three elements from a set of four. The number of ways to do this can be calculated using the formula for combinations: \( \binom{n}{r} \), where \(n\) is the total number of elements, and \(r\) is the number of elements to choose. Thus, the number of ways to choose three elements from a set of four is \( \binom{4}{3} = 4 \).
Hence, there are four distinct ways to select three elements from four, leading to four possible subgroups of \(S_4\) that are isomorphic to \(S_3\). This essential combinatorial reasoning helps in understanding how different substructures can exist within a larger mathematical framework.