Question

(a) The set of even permutations of four objects (a proper subgroup of \(S_{4}\) ) is known as the alternating group \(A_{4-}\) List its twelve members using cycle notation. (b) Assume that all permutations with the same cycle structure belong to the same conjugacy class. Show that this leads to a contradiction and hence demonstrates that even if two permutations have the same cycle structure they do not necessarily belong to the same class. (c) By evaluating the products \(p_{1}=(123)(4) \cdot(12)(34) \cdot(132)(4)\) and \(p_{2}=\) \((132)(4) \cdot(12)(34) \cdot(123)(4)\) deduce that the three elements of \(A_{4}\) with structure of the form (12)(34) belong to the same class. (d) By evaluating products of the form \((1 \alpha)(\beta \gamma) \bullet(123)(4) \bullet(1 \alpha)(\beta \gamma)\), where \(\alpha, \beta, \gamma\) are various combinations of \(2,3,4\), show that the class to which \((123)(4)\) belongs contains at least four members. Show the same for \((124)(3)\). (e) By combining results (b), (c) and (d) deduce that \(A_{4}\) has exactly four classes, and determine the dimensions of its irreps. (f) Using the orthogonality properties of characters and noting that elements of the form \((124)\) (3) have order 3, find the character table for \(A_{4}\)

Short answer

A4 has 12 elements. Permutations with the same cycle structure don't always belong to the same class, but (12)(34) conforms to a single class. A4 has 4 classes.

Step-by-step solution

List all 12 even permutations of four objects (part a)

The even permutations can be listed using cycle notation: (1), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143), (234), (243)

Verify Conjugacy Class (part b)

Permutations with the same cycle structure do not always belong to the same conjugacy class. For example, (12)(34) and (13)(24) have the same cycle structure but cannot be transformed into each other using a permutation.

Evaluate Products of Permutations (part c)

Evaluate product: \[ p_{1}=(123)(4) \cdot (12)(34) \cdot (132)(4) \] and \[ p_{2}=(132)(4) \cdot (12)(34) \cdot (123)(4) \] Both results show that the elements with the same cycle structure (12)(34) belong to the same class.

Evaluate Products to show class members (part d)

Evaluate \[ (1 \alpha)(\beta \gamma) \cdot (123)(4) \cdot (1 \alpha)(\beta \gamma) \] for various combinations of \( \alpha, \beta, \gamma \). This shows that the class to which (123)(4) belongs contains at least four members. Proceed similarly for (124)(3).

Deduce exact classes in A4 (part e)

Using observations from steps b, c, and d conclude that \(A_{4}\) has exactly four classes. Determine the dimensions of irreducible representations (irreps).

Construct character table (part f)

Using the orthogonality properties of characters and knowing element orders, construct the character table for \( A_{4} \). For example, forms (124)(3) have an order of 3.

Key concepts

Group Theory

In mathematics, group theory is the study of algebraic structures known as 'groups'. A group is a collection of elements combined with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. Group theory is pivotal in understanding the symmetry and structure in various mathematical contexts.
For the alternating group \(A_4\), which consists of even permutations of four objects, understanding its group properties helps in dissecting its structure and behavior. \(A_4\) is a subgroup of \(S_4\), the symmetric group of four elements, and includes 12 unique even permutations. These permutations can be visualized and studied using 'cycle notation'.
Understanding these basics lays the groundwork for exploring deeper concepts like conjugacy classes and irreducible representations.

Cycle Notation

Cycle notation is a method to express permutations by denoting the cyclic structure of the permutation. Instead of listing all elements, we capture the effect of the permutation in cycles.
For instance, the permutation \( (123) \) implies that 1 goes to the position of 2, 2 goes to the position of 3, and 3 returns to the position of 1. In the context of \(A_4\), the 12 even permutations can be written in cycle notation as:

• (1)
• (12)(34)
• (13)(24)
• (14)(23)
• (123)
• (132)
• (124)
• (142)
• (134)
• (143)
• (234)
• (243)

Using this notation simplifies the process of identifying and working with permutations in larger groups like \(A_4\). Cycle notation is also key in understanding conjugacy classes and product evaluations within the group.

Conjugacy Classes

In group theory, a conjugacy class is a set of elements that are conjugate to each other. Two elements \(a\) and \(b\) of a group \(G\) are conjugate if there exists an element \(g \in G\) such that \( b = gag^{-1} \). This relation partitions the group into disjoint subsets called conjugacy classes.
In \(A_4\), it is critical to recognize that elements having the same cycle structure do not necessarily belong to the same conjugacy class. For example, (12)(34) and (13)(24) have the same two-cycle structure but belong to different conjugacy classes. This is crucial for accurately identifying the group's structure and the relationships within its elements.
Evaluating products within \(A_4\) helps verify these conjugacy classes. For instance, evaluating certain products verifies that all elements of the form (12)(34) belong to the same class. Similarly, analyzing products like \( (1α)(βγ) \bullet (123)(4) \bullet (1α)(βγ) \) aids in identifying class memberships accurately.

Irreducible Representations

Irreducible representations (irreps) are a fundamental concept in group theory, providing insight into how a group can be represented as linear transformations of vector spaces. An irrep is a representation that cannot be decomposed into smaller, non-trivial representations.
For \(A_4\), knowing its conjugacy classes allows us to determine the dimensions of its irreducible representations. By examining the observations made in steps like evaluating permutation products, it becomes evident that \(A_4\) has exactly four irreps. The dimensions of these irreps can be deduced from the group's structure and representation theory principles.
These irreducible representations are crucial for constructing the character table of \(A_4\), which encapsulates significant information about the group's behavior in various contexts.

Character Table

A character table is a compact way of encapsulating the character values of a group’s irreducible representations over its conjugacy classes. Characters are essentially traces of the matrices representing group actions in each irrep.
For the alternating group \(A_4\), constructing the character table involves:

• Identifying the conjugacy classes.
• Determining the irreps and their dimensions.
• Using orthogonality properties of characters.

Knowing specific details, like elements of the form (124) having order 3, helps in filling out the character table. The orthogonality relations ensure that the computed table is consistent and accurate.
A typical character table for \(A_4\) might look like:
\begin{array}{c|cccc}\ & C_1 & C_2 & C_3 & C_4 \hline\chi_1 & 1 & 1 & 1 & 1 \chi_2 & 1 & -1 & 1 & -1 \chi_3 & 2 & 0 & -1 & 0 \chi_4 & 3 & 0 & 0 & -1 \end{array}
This table is a powerful tool in group theory, providing a look into the structure and properties of \(A_4\).