Question

Determine whether the indicated subgroup is normal in the indicated group. $$ \langle(123)\rangle \text { in } S_{4} $$

Short answer

\(\langle(123)\rangle\) is not a normal subgroup of \(S_4\).

Step-by-step solution

Understand the Group Structure

The symmetric group, \( S_4 \), is the group of all permutations of four elements. It has \( 4! = 24 \) elements. This is a non-abelian group since not all permutations commute.

Define the Subgroup

The subgroup \( \langle (123) \rangle \) is generated by the cycle \((123)\), meaning it consists of all powers of \((123)\). The elements of this subgroup are \( \{e, (123), (132) \} \), where \( e \) is the identity element.

Check for Normality

A subgroup \( H \) of a group \( G \) is normal if for every \( g \in G \) and \( h \in H \), the conjugate \( g h g^{-1} \) is also in \( H \). Thus, we need to verify whether \( g(123)g^{-1} \in \{e, (123), (132)\} \) for all \( g \in S_4 \).

Calculate Conjugates

Take elements of \( S_4 \) such as \((12), (13)\), and others, and calculate their conjugates: \((12)(123)(12) = (132)\). Since \( (132) \in \langle(123)\rangle \), this specific conjugate is in the subgroup. We must check similarly with other elements of \( S_4 \).

General Verification

Verify the conjugacy condition for a variety of elements, ensuring that \( g(123)g^{-1} \) results in some element in \( \{e, (123), (132)\} \). If any conjugate of \((123)\) exits the subgroup, \( \langle(123)\rangle \) is not normal.

Conclusion

Upon detailed checking, find that not all conjugates of elements of \( \langle(123)\rangle \) remain within the subgroup for every \( g \in S_4 \). For instance, some conjugation operations yield results not contained in \( \langle(123)\rangle \), confirming that this subgroup is not normal.

Key concepts

Symmetric Group

In mathematics, a symmetric group is a fundamental structure that plays a critical role in the study of permutations. The symmetric group on four elements, denoted as \( S_4 \), is the group containing all possible permutations of the four elements. This means \( S_4 \) has 24 distinct permutations since there are \( 4! \) (4 factorial) different ways to arrange four numbers. These properties make the symmetric group a non-abelian group, meaning that not all permutations commute; in other words, the order in which permutations are applied impacts the result.

A group is a set equipped with an operation that combines any two of its elements to form a third element while satisfying four conditions: closure, associativity, identity, and invertibility. In \( S_4 \), the operation is function composition.

• Closure: The composition of any two permutations is also a permutation in \( S_4 \).
• Associativity: The order of applying permutations doesn’t matter within a sequence of compositions.
• Identity: The identity permutation, leaving all elements in their original positions, exists in \( S_4 \).
• Invertibility: Every permutation has an inverse that returns the set to its original order.

Permutation

A permutation is a rearrangement of elements in a particular set. In the context of the symmetric group \( S_4 \), permutations are the rearrangements of four elements. Each arrangement or reordering counts as a distinct permutation. A permutation can be represented in index notation or cycle notation.

Cycle notation is a powerful tool to describe permutations succinctly. For example, the cycle \((123)\) represents the permutation that moves element 1 to the position of element 2, element 2 to the position of element 3, and element 3 back to the position of element 1. The remaining element(s), if any, remain fixed or unchanged, often left out of the cycle notation unless necessary.

• Permutations are essential because they describe the idea that any set can be rearranged in multiple ways.
• Performing permutations involves understanding and correctly applying these rearrangements.

Conjugate

In group theory, conjugates are a significant concept related to understanding the internal structure of groups. Given a group \( G \) and elements \( g, h \in G \), the conjugate of \( h \) by \( g \) is expressed as \( g h g^{-1} \). This expression indicates a transformation of \( h \) by \( g \).

Conjugation involvement in groups helps determine normal subgroups. A key property of conjugates is that if \( h \) is conjugated by any element in the group and still results in an element from the subgroup, then the subgroup is potentially normal.

• Conjugates help analyze whether a subgroup behaves uniformly under the group's operation.
• They provide insights into the symmetry properties of the group.

The focus in this context is ensuring conjugation leaves the group structure within the expected subgroup boundaries.

Subgroup

A subgroup is a subset of a group that is itself a group under the operation defined on the original group. Determining if a subset is a subgroup involves checking that it satisfies the group properties: closure, associativity, identity, and invertibility.

In the example of the symmetric group \( S_4 \), the particular subgroup of interest is generated by the cycle \((123)\), denoted as \( \langle (123) \rangle \). This means it contains all the powers of \((123)\). The elements \( \{ e, (123), (132) \} \) make up this subgroup, where \( e \) is the identity element, \( (123) \) is the cycle itself, and \( (132) \) is its inverse cycle.

• Subgroups retain the original group's structure but are contained within a broader group.
• Not all subgroups in a group are necessarily normal subgroups.

Understanding subgroups enhances comprehension of the group's internal configuration.

Cycle Notation

Cycle notation is a concise way of representing permutations and is instrumental in working with symmetric groups. Rather than listing the result of each element's transformation individually, cycle notation groups elements that are permuted among themselves.

For instance, the cycle \((123)\) indicates that element 1 moves to the position of element 2, 2 moves to 3, and 3 moves back to 1, effectively forming a closed loop. Longer permutations can be split into several cycles, which are called disjoint cycles. Cycle notation provides simplicity and clarity in expressing complex permutations.

• Cycle notation is succinct and highlights key interactions in permutations.
• It displays permutations as products of independent cycles, aiding in visualization.

This representation is crucial for calculating powers, conjugates, and understanding the group's overall structure in permutation studies.