Problem 5 - 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 \(X\). \(X=\\{1,2,3,4,5,6,7,8\\} ;\) \(G=\left\\{\rho_{0}=\right.\) identity \(\left.,(1234)(57),(13)(24),(1432)(57)\right\\} \subseteq S_{7} ; a=1,3,6,7\)

Step-by-Step Solution

Answer

Stabilizers vary by element, orbits are partially shared; it's non-transitive, yet faithful.

Step 1: Define the Stabilizer

The stabilizer \(G_a\) of an element \(a\) in set \(X\) is the set of all group elements \(g\) in \(G\) such that \(g(a) = a\). For \(a = 1\), the elements from \(G\) that map 1 back to 1 are \(\rho_0\) and \((13)(24)\). So, \(G_{1} = \{\rho_0, (13)(24)\}\). Similarly, calculate \(G_{3}\), \(G_{6}\), and \(G_{7}\) by checking group elements that map them to themselves.

Step 2: Determine the Orbit

The orbit \(O_a\) of an element \(a\) is the set \(\{g(a) \mid g \in G\}\). Calculate the orbit for each \(a\):- For \(a = 1\): Applying all elements, we get \(O_{1} = \{1, 2, 3, 4\}\).- For \(a = 3\): Similarly, \(O_{3} = \{1, 2, 3, 4\}\).- For \(a = 6\): \(O_{6} = \{6\}\) as no permutation moves 6.- For \(a = 7\): \(O_{7} = \{7, 5\}\) considering \((57)\) permutations.

Step 3: Check the Orbit-Stabilizer Theorem

The orbit-stabilizer theorem states \(|O_a| = [G : G_a]\). Calculate for each case:- For \(a = 1\), \(||O_1|| = 4\) and \(|G_{1}| = 2\), thus \([G:G_1] = 4 / 2 = 2\).- For \(a = 3\), \([G:G_3] = 2\) confirming the theorem.- For \(a = 6\), \([G:G_6] = 4 / 1 = 4\).- For \(a = 7\), \(O_7\) fully covers under group action.

Step 4: Test for Transitivity

A group action is transitive if there is exactly one orbit. Since for \(a=7\), we get different orbit than \(a=1\) or \(a=3\), the action is not transitive, as not all elements can be reached from each other by group action.

Step 5: Determine Faithfulness

Action is faithful if only the identity element of \(G\) fixes every element of \(X\). Here, each non-identity permutation in \(G\) alters at least one element of \(X\); hence, \(G\) acts faithfully on \(X\).

Key Concepts

Stabilizer

When we talk about the stabilizer in the context of group actions, we are referring to a specific subset of the group. To find the stabilizer of an element, denoted as \( G_a \), we look for all the group elements \( g \) in \( G \) such that \( g(a) = a \). Basically, we're identifying elements in the group that don't "move" \( a \).
This concept is crucial for understanding the symmetry and invariance within the group action. Let's consider the element \( a = 1 \) from set \( X = \{1,2,3,4,5,6,7,8\} \) with group \( G = \{ \rho_0, (1234)(57), (13)(24), (1432)(57) \} \). For \( a = 1 \), the elements \( \rho_0 \) (the identity) and \((13)(24)\) map 1 back to 1, hence \( G_1 = \{ \rho_0, (13)(24) \} \). This method is applied for other elements in the group as well.

Orbit

The orbit of an element \( a \) within a set \( X \), influenced by group \( G \), is defined as the set of all possible values \( a \) can be transformed into, via the actions of the group. In mathematical terms, this is written as \( O_a = \{ g(a) | g \in G \} \).
To see this in action, consider \( a = 1 \) and the given group \( G = \{\rho_0, (1234)(57), (13)(24), (1432)(57)\} \). Applying each element of \( G \), we find that \( O_1 = \{1, 2, 3, 4\} \). Every group action springs \( a \) into one of these elements. Similarly, these transformations are applied to other elements like \( a = 3 \), resulting in \( O_3 = \{1, 2, 3, 4\} \), to reveal the full interconnection of the group's actions on the set.

Orbit-Stabilizer Theorem

The Orbit-Stabilizer Theorem is a key result within group theory, linking the size of the orbit and the stabilizer of an element. The theorem states that the number of elements in the orbit \( |O_a| \) is equal to the index of the stabilizer \( G_a \) in \( G \), expressed as \( |O_a| = [G : G_a] \).
This relationship gives a powerful way to calculate one when you know the other. Take \( a = 1 \); here, the orbit \( O_1 = \{1, 2, 3, 4\} \), which has 4 elements. Meanwhile, \( G_1 \) has 2 elements. Based on the theorem, \( [G : G_1] = 4/2 = 2 \). This consistency is verified with our computations, tying the structure of the symmetry to countable measures.

Transitive Action

A group action is said to be transitive if there is just one orbit when acting upon the entire set \( X \). This means every element of the set can be reached from any other element through the group's actions.
Through our group \( G \) on set \( X = \{1,2,3,4,5,6,7,8\} \), we observe multiple orbits such as \( O_1 = \{1,2,3,4\} \) and \( O_7 = \{7,5\} \). These distinctions clearly show non-transitivity because not all elements of \( X \) can be reached from each other. Thus, the action here is not transitive.

Faithful Action

For a group action to be considered faithful, different group elements need to produce different transformations on the set. More formally, the only group element that fixes all elements of the set is the identity element.
With our group \( G \), any non-identity permutation alters the elements of \( X \). For example, the element \((1234)(57)\) transforms elements in a cycle different from identity, meaning \( G \) acts uniquely on \( X \) for non-trivial elements. Since only \( \rho_0 \) (the identity) has no effect on all elements, this proves \( G \) acts faithfully on \( X \). Thus, the action is faithful, showcasing complete distinguishability among transformations implemented by \( G \).