Problem 4 - Algebra: Pure and Applied

Question

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). $$ X=\\{1,2,3,4\\} ; G=A_{4} \subseteq S_{4} ; a=1,3,4 $$

Step-by-Step Solution

Answer

\(A_4\) acts transitively and faithfully on \(X = \{1, 2, 3, 4\}\), stabilizers are \(\{(1), (23)(24), (24)(23)\}\) per element.

Step 1: Understanding the Symmetric Group and Alternating Group

First, it's important to know that the symmetric group, \(S_4\), consists of all permutations of the set \(X = \{1, 2, 3, 4\}\). The alternating group, \(A_4\), is the subgroup of \(S_4\) consisting of even permutations, or those that can be written as an even number of transpositions.

Step 2: Finding the Stabilizer for Each Element

The stabilizer, \(G_a\), of an element \(a\) in a group action is the subgroup of \(G\) that keeps \(a\) fixed. For each \(a \in X\), find the elements of \(A_4\) that fix \(a\):* \(a = 1\): The permutations fixing 1 are (1), (23)(24), (24)(23). So, \(|G_1|=3\).* \(a = 3\): Similar reasoning gives us (1), (12)(34), (14)(23). So, \(|G_3|=3\).* \(a = 4\): Find permutations that leave 4 fixed like (1), (13)(24), and (12)(34). So, \(|G_4|=3\).

Step 3: Finding the Orbit of Each Element

The orbit, \(O_{a}\), of an element \(a\) is the set of elements that \(a\) can be sent to by the elements of \(G\). Since \(A_4\) acts transitively on its orbits and the order \(|A_4| = 12\), orbits are calculated as:* \(O_1 = \{1, 2, 3, 4\}\) since transitive action.* \(O_3 = \{1, 2, 3, 4\}\)* \(O_4 = \{1, 2, 3, 4\}\)

Step 4: Checking the Orbit-Stabilizer Theorem

The Orbit-Stabilizer Theorem states that \(\left|O_a\right| = \left[G : G_a\right]\), where \(\left[G : G_a\right]\) is the index of \(G_a\) in \(G\). Check this for each element:* \(|O_1| = 4\), \(|G_1| = 3\), so \(|O_1| = \frac{12}{3} = 4\)* \(|O_3| = 4\), \(|G_3| = 3\), so \(|O_3| = \frac{12}{3} = 4\)* \(|O_4| = 4\), \(|G_4| = 3\), so \(|O_4| = \frac{12}{3} = 4\)The relation holds in all cases.

Step 5: Determining the Transitivity of the Group Action

An action is transitive if there is a single orbit for the entire set. Since for all elements \(a\), \(|O_a| = 4\), \(A_4\) acts transitively on \(X\).

Step 6: Checking If the Action Is Faithful

An action is faithful if the only group element that fixes every element of \(X\) is the identity. Since \(A_4\) acts without any non-identity elements fixing all \(\{1, 2, 3, 4\}\), \(A_4\) acts faithfully on \(X\).

Key Concepts

Symmetric Group

The symmetric group, denoted as \(S_n\), is a fundamental concept in group theory. It consists of all possible permutations of a set with \(n\) elements. Permutations are essentially ways of rearranging elements. For instance, in our exercise, \(S_4\) includes all permutations of the set \(X = \{1,2,3,4\}\). This means any way of changing the order of 1, 2, 3, and 4 is in \(S_4\).

To represent permutations, we use special notation to show how each element is exchanged. For example, the permutation \((1 ext{ ext{ } 2 ext{ ext{ } 3 ext{ ext{ } 4})}\) means that 1 stays as 1, 2 becomes 2, and so on, which is effectively doing nothing, hence it is called the identity permutation.

Understanding \(S_n\) is crucial because it forms the backbone of more complex groups, like the alternating group.

Alternating Group

The alternating group, denoted as \(A_n\), is a subgroup of the symmetric group \(S_n\). This group is composed solely of even permutations. An even permutation is one that can be expressed as an even number of transpositions, where a transposition swaps exactly two elements.

For example, in \(A_4\), which is a subset of \(S_4\), only those permutations that can be made with an even number of swaps are included. If we consider the permutation \((12)(34)\), it swaps 1 with 2 and 3 with 4, making it an even permutation since it consists of two swaps.

The order of \(A_n\) is exactly half of the order of \(S_n\), which makes sense because half of the permutations are even, while the other half are odd. Hence, for \(A_4\), the order is 12 since \(S_4\) has 24 permutations.

The alternating group is important because it is the smallest non-trivial normal subgroup of the symmetric group.

Stabilizer

In group theory, the stabilizer of an element, denoted as \(G_a\), is a vital concept when analyzing group actions. The stabilizer is the subset of the group that leaves an element \(a\) unchanged. For our set \(X = \{1, 2, 3, 4\}\), this means finding permutations in \(A_4\) that do not change the position of an element when the permutation is applied.

So, if we consider \(a = 1\), the stabilizer \(G_1\) includes permutations like the identity \((1)\) and swaps that do not move 1's position, such as \((23)(24)\) and \((24)(23)\).

The order of a stabilizer group provides information on how restrictive the action is when fixing an element. The stabilizer's size helps in calculating other properties like orbits, using concepts such as the orbit-stabilizer theorem.

Orbit

The orbit of an element \(a\), denoted as \(O_a\), within a group, is the set of all elements that an element \(a\) can be transformed into using the group actions. This gives a pathway showing where \(a\) can "travel" under the actions of group elements. For the set \(X = \{1, 2, 3, 4\}\) and the group \(A_4\), the orbit \(O_1\) for \(a = 1\) consists of all elements in \(X\) because \(A_4\) acts transitively, meaning all elements can reach any other.

This transitively great spread of orbits confirms the powerful reach of our group action, further reflected in the Orbit-Stabilizer theorem. The theorem mathematically expresses this connection by stating the size of the orbit times the size of the stabilizer equals the total number of group actions.

Understanding orbits is essential for determining if a group action is transitive, which in turn influences how a group acts on sets and their elements.